Exponential and
Logarithmic Functions
1 Exponential Functions; Continuous Compounding
2 Logarithmic Functions
3 Differentiation of Exponential and Logarithmic Functions
4 Applications; Exponential Models
Chapter Summary
Important Terms, Symbols, and Formulas
Checkup for Chapter 4
Review Exercises
Explore! Update
Think About It
Exponential functions can be used for describing the effect of
medication.
4
291
CHAPTER4
SECTION 4.1 Exponential Functions; Continuous Compounding
A population Q(t) is said to grow exponentially if whenever it is measured at equally
spaced time intervals, the population at the end of any particular interval is a fixed
multiple (greater than 1) of the population at the end of the previous interval. For
instance, according to the United Nations, in the year 2000, the population of the
world was 6.1 billion people and was growing at an annual rate of about 1.4%. If this
pattern were to continue, then every year, the population would be 1.014 times the
population of the previous year. Thus, if P(t) is the world population (in billions) t
years after the base year 2000, the population would grow as follows:
2000 P(0) 6.1
2001 P(1) 6.1(1.014) 6.185
2002 P(2) 6.185(1.014) [6.1(1.014)](1.014) 6.1(1.014)2 6.272
2003 P(3) 6.272(1.014) [6.1(1.014)2](1.014) 6.1(1.014)3 6.360
2000 t P(t) 6.1(1.014)t
The graph of P(t) is shown in Figure 4.1a. Notice that according to this model,
the world population grows gradually at first but doubles after about 50 years (to
12.22 billion in 2050).
FIGURE 4.1 Two models for population growth.
Exponential population models are sometimes referred to as Malthusian, after
Thomas Malthus (1766–1834), a British economist who predicted mass starvation
would result if a population grows exponentially while the food supply grows at a constant
rate (linearly). Fortunately, world population does not continue to grow exponentially
as predicted by Malthus’s model, and models that take into account various
restrictions on the growth rate actually provide more accurate predictions. The population
curve that results from one such model, the so-called logistic model, is shown in
Figure 4.1b. Note how the logistic growth curve rises steeply at first, like an exponential
curve, but then eventually turns over and flattens out as environmental factors act
to brake the growth rate. We will examine logistic curves in Section 4.4 (Example 4.4.6)
and again in Chapter 6 as part of a more detailed study of population models.
A function of the general form f (x) bx, where b is a positive number, is called
an exponential function. Such functions can be used to describe exponential and
t
y
0
y L(t)
(b) A logistic population curve
t
y
6.1
10
0
(2000)
y P(t)
(a) The graph of the exponential
growth function P(t) 6.1(1.014)t
20
50
(2050)
292 CHAPTER 4 Exponential and Logarithmic Functions 4-2
logistic growth and a variety of other important quantities. For instance, exponential
functions are used in demography to forecast population size, in finance to calculate
the value of investments, in archaeology to date ancient artifacts, in psychology to
study learning patterns, and in industry to estimate the reliability of products.
In this section, we will explore the properties of exponential functions and introduce
a few basic models in which such functions play a prominent role. Additional
applications such as the logistic model are examined in subsequent sections.
Working with exponential functions requires the use of exponential notation
and the algebraic laws of exponents. Solved examples and practice problems
involving this notation can be found in Appendix A1. Here is a brief summary of
the notation.
4-3 SECTION 4.1 EXPONENTIAL FUNCTIONS; CONTINUOUS COMPOUNDING 293
Definition of bn for Rational Values of n (and b . 0) ■ Integer powers:
If n is a positive integer,
bn b b b
Fractional powers: If n and m are positive integers,
where denotes the positive mth root.
Negative powers:
Zero power: b0 1
b n
1
bn
m
b
bn/m ( m b)n
m
bn
afddfdbfdddfc
n factors
For example,
34 3 3 3 3 81
We know what is meant by br for any rational number r, but if we try to graph
y bx there will be a “hole” in the graph for each value of x that is not rational, such
as . However, using methods beyond the scope of this book, it can be shown
that irrational numbers can be approximated by rational numbers, which in turn implies
there is only one unbroken curve passing through all points (r, br) for r rational. In other
words, there exists a unique continuous function f(x) that is defined for all real numbers
x and is equal to br when r is rational. It is this function we define as f (x) br.
Exponential Functions ■ If b is a positive number other than 1 (b 0,
b 1), there is a unique function called the exponential function with base b that
is defined by
f (x) bx for every real number x
x 2
27 2/3
1
( 3 27)2
1
32
1
9
4 3/2
1
43/2
1
8
41/2 43/2 ( 4)3 23 8 4 2
3 4
1
34
1
81
EXPLORE!
Graph y ( 1)x using a
modified decimal window
[ 4.7, 4.7]1 by [ 1, 7]1.
Why does the graph appear
with dotted points? Next
set b to decimal values,
b 0.5, 0.25, and 0.1, and
graph y bx for each case.
Explain the behavior of
these graphs.
To get a feeling for the appearance of the graph of an exponential function, consider
Example 4.1.1.
EXAMPLE 4.1.1
Sketch the graphs of y 2x and y .
Solution
Begin by constructing a table of values for y 2x and y : 1
2 x
1
2 x
294 CHAPTER 4 Exponential and Logarithmic Functions 4-4
y 2x 0.00003 0.001 0.5 1 2 8 32 1,024 32,768
y 32,768 1,024 2 1 0.5 0.125 0.313 0.001 0.00003 1
2 x
x 15 10 1 0 1 3 5 10 15
The pattern of values in this table suggests that the functions y 2x and y
have the following features:
Using this information, we sketch the graphs shown in Figure 4.2. Notice that each
graph has (0, 1) as its y intercept, has the x axis as a horizontal asymptote, and appears
to be concave upward for all x. The graphs also appear to be reflections of one another
in the y axis. You are asked to verify this observation in Exercise 74.
FIGURE 4.2 The graphs of y 2x and .
Figure 4.3 shows graphs of various members of the family of exponential functions
y bx. Notice that the graph of any function of the form y bx resembles that
of y 2x if b 1 or y if 0 b 1. In the special case where b 1, the
function y bx becomes the constant function y 1.
1
2 x
y 1
2 x
x
y
y 2x
(0, 1)
0
y 1
2 x
lim
x→ 1
2 x
lim 0
x→
2x
lim
x→ 1
2 x
lim ∞
x→
2x 0
The function y 1
2 x
always decreasing
The function y 2x
always increasing
1
2 x
EXPLORE!
Graph y bx, for b 1, 2, 3,
and 4, using the modified
decimal window [ 4.7, 4.7]1
by [ 1, 7]1. Explain what you
observe. Conjecture and then
check where the graph of
y 4x will lie relative to that
of y 2x and y 6x. Where
does y ex lie, assuming e is
a value between 2 and 3?
FIGURE 4.3 Graphs of the exponential form y bx.
Important graphical and analytical properties of exponential functions are summarized
in the following box.
x
y
y 3x
y 2x
y (0.7)x y 1.3x
y 1
0
y 1
2 x
y 1
3 x
4-5 SECTION 4.1 EXPONENTIAL FUNCTIONS; CONTINUOUS COMPOUNDING 295
Properties of an Exponential Function ■ The exponential function
for b 0, b
1 has these properties:
1. It is defined, continuous, and positive (bx 0) for all x
2. The x axis is a horizontal asymptote of the graph of f.
3. The y intercept of the graph is (0, 1); there is no x intercept.
4. If b 1, and
If 0 b 1, and
5. For all x, the function is increasing (graph rising) if b 1 and decreasing
(graph falling) if 0 b l.
lim
x→
lim bx 0.
x→
bx
lim
x→
lim bx .
x→
bx 0
f (x) bx
NOTE Students often confuse the power function p(x) xb with the exponential
function f (x) bx. Remember that in xb, the variable x is the base and the
exponent b is constant, while in bx, the base b is constant and the variable x is
the exponent. The graphs of y x2 and y 2x are shown in Figure 4.4. Notice
that after the crossover point (4, 16), the exponential curve y 2x rises much
more steeply than the power curve y x2. For instance, when x 10, the y value
on the power curve is y 102 100, while the corresponding y value on the
exponential curve is y 210 1,024. ■
FIGURE 4.4 Comparing the power curve y x2 with the exponential curve y 2x.
x
y
y 2x y x2
(2, 4)
(4, 16)
0.77 2 4
EXAMPLE 4.1.2
Evaluate each of these exponential expressions:
a. (3)2(3)3 b. (23)2 c.
d. e.
Solution
a. (3)2(3)3 32+3 35 243
b. (23)2 2(3)(2) 26 64
c. (51/3)(21/3) [(5)(2)]1/3 101/3
d.
e.
EXAMPLE 4.1.3
If , find all values of x such that f (x) 125.
Solution
The equation f (x) 125 53 is satisfied if and only if
x2 2x 3 since bx by only when x y
x2 2x 3 0
(x 1)(x 3) 0 factor
x 1, x 3
Thus, f (x) 125 if and only if x 1 or x 3.
5x2 2x 53
f(x) 5x2 2x
4
7 3
43
73
64
343
23
25 23 5 2 2
1
4
3 10
4
7 23 3
25
(51/3)(21/3)
296 CHAPTER 4 Exponential and Logarithmic Functions 4-6
Exponential Rules ■ For bases a, b (a 0, b 0) and any real numbers
x, y, we have
The equality rule: bx by if and only if x y
The product rule: bxby bx y
The quotient rule:
The power rule: (bx ) y bxy
The multiplication rule: (ab)x axbx
The division rule: a
b x
ax
bx
bx
by bx y
Exponential functions obey the same algebraic rules as the rules for exponential
numbers reviewed in Appendix A1. These rules are summarized in the following box.
In algebra, it is common practice to use the base b 10 for exponential functions or,
in some cases, b 2, but in calculus, it turns out to be more convenient to use a number
denoted by e and defined by the limit
“Hold on!” you say, “That limit has to be 1, since certainly tends to 1 as
n increases without bound, and 1n 1 for any n.” Not so. The limit process does not
work this way, as you can see from this table:
1
1
n
e lim
n→
1
1
n n
The Natural
Exponential Base e
4-7 SECTION 4.1 EXPONENTIAL FUNCTIONS; CONTINUOUS COMPOUNDING 297
The number e is one of the most important numbers in all mathematics,
and its value has been computed with great precision. To twelve decimal places,
its value is
The function is called the natural exponential function. To compute eN
for a particular number N, you can use a table of exponential values or, more likely,
the “eX” key on your calculator. For example, to find e1.217 press the eX key and then
enter the number 1.217 to obtain e1.217 3.37704.
EXAMPLE 4.1.4
A manufacturer estimates that when x units of a particular commodity are produced,
they can all be sold when the market price is p dollars per unit, where p is given by
the demand function p 200e 0.01x. How much revenue is obtained when 100 units
of the commodity are produced?
Solution
The revenue is given by the product (price/unit)(number of units sold); that is,
Using a calculator, we find that the revenue obtained by producing x 100 units is
or approximately $7,357.59.
EXAMPLE 4.1.5
Biologists have determined that the number of bacteria in a culture is given by
P(t) 5,000e0.015t
R(100) 200(100) e 0.01(100) 7357.59
R(x) p(x)x (200e 0.01x)x 200xe 0.01x
y ex
e lim
n→
1
1
n n
2.718281828459 . . .
EXPLORE!
Store into Y1 of the
function editor and examine its
graph. Trace the graph to the
right for large values of x. What
number is y approaching as x
gets larger and larger? Try
using the table feature of the
graphing calculator, setting both
the initial value and the
incremental change first to 10
and then successively to 1,000
and 100,000. Estimate the limit
to five decimal places. Now do
the same as x approaches
and observe this limit.
1
1
x x
n 10 100 1,000 10,000 100,000 1,000,000
1 2.59374 2.70481 2.71692 2.71815 2.71827 2.71828
1
n n
where t is the number of minutes after observation begins. What is the average rate
of change of the bacterial population during the second hour?
Solution
During the second hour (from time t 60 to t 120), the population changes by
P(120) P(60), so the average rate of change during this time period is given by
Thus, the population increases at the average rate of roughly 299 bacteria per minute
during the second hour.
The number e is called the “natural exponential base,” but it may seem anything but
“natural” to you. As an illustration of how this number appears in practical situations,
we use it to describe the accounting practice known as continuous compounding of
interest.
First, let us review the basic ideas behind compound interest. Suppose a sum of
money is invested and the interest is compounded only once. If P is the initial investment
(the principal) and r is the interest rate (expressed as a decimal), the balance B
after the interest is added will be
B P Pr P(1 r) dollars
That is, to compute the balance at the end of an interest period, you multiply the balance
at the beginning of the period by the expression 1 r.
At most banks, interest is compounded more than once a year. The interest that
is added to the account during one period will itself earn interest during the subsequent
periods. If the annual interest rate is r and interest is compounded k times per
year, then the year is divided into k equal compounding periods and the interest rate
in each period is . Hence, the balance at the end of the first period is
principal interest
At the end of the second period, the balance is
P 1
r
k 1
r
k P 1
r
k 2
P2 P1 P1 r
k P1 1
r
k
P1 P P r
k P 1
r
k
r
k
Continuous
Compounding
of Interest
299
30,248 12,298
60
[5,000e0.015(120)] [5,000e0.015(60)]
60
A
P(120) P(60)
120 60
298 CHAPTER 4 Exponential and Logarithmic Functions 4-8
→
→ abc adddddbdddddc
and, in general, the balance at the end of the mth period is
Since there are k periods in a year, the balance after 1 year is
At the end of t years, interest has been compounded kt times and the balance is given
by the function
As the frequency with which interest is compounded increases, the corresponding
balance B(t) also increases. Hence, a bank that compounds interest frequently
may attract more customers than one that offers the same interest rate but
compounds interest less often. But what happens to the balance at the end of t
years as the compounding frequency increases without bound? More specifically,
what will the balance be at the end of t years if interest is compounded not quarterly,
not monthly, not daily, but continuously? In mathematical terms, this
question is equivalent to asking what happens to the expression as k
increases without bound. The answer turns out to involve the number e. Here is the
argument.
To simplify the calculation, let . Then, k nr and so
Since n increases without bound as k does, and since approaches e as n
increases without bound, it follows that the balance after t years is
To summarize:
B(t) lim
k→
P 1
r
k kt
P lim
n→ 1
1
n n rt
Pert
1
1
n n
P 1
r
k kt
P 1
1
n nrt
P 1
1
n n rt
n
k
r
P 1
r
k kt
B(t) P 1
r
k kt
P 1
r
k k
Pm P 1
r
k m
4-9 SECTION 4.1 EXPONENTIAL FUNCTIONS; CONTINUOUS COMPOUNDING 299
EXPLORE!
Suppose you have $1,000 to
invest. Which is the better
investment, 5% compounded
monthly for 10 years or 6%
compounded quarterly for 10
years? Write the expression
1,000(1 R K)^(K*T) on the
Home Screen and evaluate
after storing appropriate
values for R, K, and T.
Compound Interest Formulas ■ Suppose P dollars are invested at an
annual interest rate r and the accumulated value (called future value) in the account
after t years is B(t) dollars. If interest is compounded k times per year, then
and if interest is compounded continuously
B(t) Pert
B(t) P 1
r
k kt
EXAMPLE 4.1.6
Suppose $1,000 is invested at an annual interest rate of 6%. Compute the balance
after 10 years if the interest is compounded
a. Quarterly b. Monthly c. Daily d. Continuously
Solution
a. To compute the balance after 10 years if the interest is compounded quarterly,
use the formula , with t 10, P 1,000, r 0.06, and k 4:
b. This time, take t 10, P 1,000, r 0.06, and k 12 to get
c. Take t 10, P 1,000, r 0.06, and k 365 to obtain
d. For continuously compounded interest use the formula B(t) Pert, with t 10,
P 1,000, and r 0.06:
B(10) 1,000e0.6 $1,822.12
This value, $1,822.12, is an upper bound for the possible balance. No matter how
often interest is compounded, $1,000 invested at an annual interest rate of 6% cannot
grow to more than $1,822.12 in 10 years.
In many situations, it is useful to know how much money P must be invested at a
fixed compound interest rate in order to obtain a desired accumulated (future) value
B over a given period of time T. This investment P is called the present value of the
amount B to be received in T years. Present value may be regarded as a measure
of the current worth of an investment and is used by economists to compare different
investment possibilities.
To derive a formula for present value, we need only solve an appropriate future
value formula for P. In particular, if the investment is compounded k times per year
at an annual rate r for the term of T years, then
and the present value of B dollars in T years is obtained by multiplying both sides of
the equation by to get
P B 1
r
k kT
1
r
k kT
B P 1
r
k kT
Present Value
B(10) 1,000 1
0.06
365 3,650
$1,822.03
B(10) 1,000 1
0.06
12 120
$1,819.40
B(10) 1,000 1
0.06
4 40
$1,814.02
B(t) P 1
r
k kt
300 CHAPTER 4 Exponential and Logarithmic Functions 4-10
Just-In-Time REVIEW
If the equation
can be solved for P
by multiplying by
to obtain
P AC 1
1
C
C 1
A PC
C
0,
EXPLORE!
Write the expressions
P*(1 R K)^(K*T)
and
P*e^(R*T)
on the Home Screen and
compare these two
expressions for the P, R, T,
and K values in Example 4.1.6.
Repeat using the same values
for P, R, and K, but with
T 15 years.
EXAMPLE 4.1.7
Sue is about to enter college. When she graduates 4 years from now, she wants to
take a trip to Europe that she estimates will cost $5,000. How much should she invest
now at 7% to have enough for the trip if interest is compounded:
a. Quarterly b. Continuously
Solution
The required future value is B $5,000 in T 4 years with r 0.07.
a. If the compounding is quarterly, then k 4 and the present value is
b. For continuous compounding, the present value is
P 5,000e 0.07(4) $3,778.92
Thus, Sue would have to invest about $9 more if interest is compounded quarterly
than if the compounding is continuous.
Which is better, an investment that earns 10% compounded quarterly, one that earns
9.95% compounded monthly, or one that earns 9.9% compounded continuously? One
way to answer this question is to determine the simple annual interest rate that is
equivalent to each investment. This is known as the effective interest rate, and it can
be easily obtained from the compound interest formulas.
Suppose interest is compounded k times per year at the annual rate r. This is
called the nominal rate of interest. Then the balance at the end of 1 year is
A P(1 i)k where i
r
k
Effective Interest
Rate
P 5,000 1
0.07
4 4(4)
$3,788.08
4-11 SECTION 4.1 EXPONENTIAL FUNCTIONS; CONTINUOUS COMPOUNDING 301
Present Value ■ The present value of B dollars in T years invested at the
annual rate r compounded k times per year is given by
If interest is compounded continuously at the same rate, the present value in
T years is given by
P Be rT
P B 1
r
k kT
EXPLORE!
Find the present value so that
the balance 25 years from
now will be $40,000 if the
annual interest rate of 6% is
compounded continuously. To
do this, place the equation
F P*e^(R*T) 0 into the
equation solver of your
graphing calculator (using the
SOLVER option), with
F 40,000, R 0.06, and
T 25. Then solve for P.
Likewise, if the compounding is continuous, then
B PerT
and the present value is given by
P Be rT
To summarize:
On the other hand, if x is the effective interest rate, the corresponding balance at the
end of 1 year is Equating the two expressions for A, we get
For continuous compounding, we have
To summarize:
Per P(1 x) so x er 1
P(1 i)k P(1 x) or x (1 i)k 1
A P(1 x).
302 CHAPTER 4 Exponential and Logarithmic Functions 4-12
Example 4.1.8 answers the question raised in the introduction to this subsection.
EXAMPLE 4.1.8
Which is better, an investment that earns l0% compounded quarterly, one that earns
9.95% compounded monthly, or one that earns 9.9% compounded continuously?
Solution
We answer the question by comparing the effective interest rates of the three investments.
For the first, the nominal rate is l0% and compounding is quarterly, so we have
and
Substituting into the formula for effective rate, we get
First effective rate (1 0.025)4 1 0.10381
For the second investment, the nominal rate is 9.95% and compounding is
monthly, so and
We find that
Second effective rate (1 0.008292)12 1 0.10417
i
r
k
0.0995
12
0.008292
r 0.0995, k 12,
i
r
k
0.10
4
0.025
r 0.10, k 4,
Effective Interest Rate Formulas ■ If interest is compounded at the nominal
rate r, the effective interest rate is the simple annual interest rate re that yields
the same interest after 1 year. If the compounding is k times per year, the effective
rate is given by the formula
while continuous compounding yields
re er 1
re (1 i)k 1 where i
r
k
EXERCISES ■ 4.1
4-13 SECTION 4.1 EXPONENTIAL FUNCTIONS; CONTINUOUS COMPOUNDING 303
In Exercises 1 and 2, use your calculator to find the
indicated power of e. (Round your answers to three
decimal places.)
1.
2.
3. Sketch the curves y 3x and y 4x on the
same set of axes.
4. Sketch the curves and
on the same set of axes.
In Exercises 5 through 12, evaluate the given
expressions.
5. a. 272/3
b.
6. a. ( 128)3/7
b.
7. a. 82/3 163/4
b.
8. a. (23 32)11/7
b. (272/3 84/3) 3/2
9. a. (33)(3 2)
b. (42/3)(22/3)
10. a.
b.
2
4/3
52
53
27 36
121 3/2
27
64 2/3 64
25 3/2
1
9 3/2
y 1
4 x
y 1
3 x
e3, e 1, e0.01, e 0.1, e2, e 1/2, e1/3, and
1
3
e
e2, e 2, e0.05, e 0.05, e0, e, e, and
1
e
11. a. (32)5/2
b. (e2e3/2)4/3
12. a.
b.
In Exercises 13 through 18, use the properties of
exponents to simplify the given expressions.
13. a. (27x6)2/3
b. (8x2y3)1/3
14. a. (x1/3)3/2
b. (x2/3) 3/4
15. a.
b. (x1.1y2)(x2 y3)0
16. a. ( 2t 3)(3t2/3)
b. (t 2/3)(t3/4)
17. a. (t5/6) 6/5
b. (t 3/2) 2/3
18. a. (x2y 3z)3
b.
In Exercises 19 through 28, find all real numbers x that
satisfy the given equation.
19. 42x 1 16
20. 3x22x 144
21. 23 x 4x
22.
23. (2.14)x 1 (2.14)1 x
4x
12
3x 8
x3y 2
z4 1/6
(x y)0
(x2y3)1/6
16
81 1/4 125
8 2/3
(31.2)(32.7)
34.1
Finally, if compounding is continuous with nominal rate 9.9%, we have r 0.099
and the effective rate is
Third effective rate e0.099 1 0.10407
The effective rates are, respectively, 10.38%, 10.42%, and 10.41%, so the second
investment is best.
24. (3.2)2x 3 (3.2)2 x
25.
26.
27.
28.
In Exercises 29 through 32, use a graphing calculator
to sketch the graph of the given exponential function.
29. y 31 x
30. y ex 2
31. y 4 e x
32. y 2x 2
In Exercises 33 and 34, find the values of the constants
C and b so that the curve y Cbx contains the
indicated points.
33. (2, 12) and (3, 24)
34. (2, 3) and (3, 9)
35. COMPOUND INTEREST Suppose $1,000
is invested at an annual interest rate of 7%.
Compute the balance after 10 years if the interest
is compounded:
a. Annually
b. Quarterly
c. Monthly
d. Continuously
36. COMPOUND INTEREST Suppose $5,000
is invested at an annual interest rate of 10%.
Compute the balance after 10 years if the interest
is compounded:
a. Annually
b. Semiannually
c. Daily (using 365 days per year)
d. Continuously
37. PRESENT VALUE How much money should be
invested today at 7% compounded quarterly so
that it will be worth $5,000 in 5 years?
38. PRESENT VALUE How much money should be
invested today at an annual interest rate of 7%
compounded continuously so that 20 years from
now it will be worth $20,000?
1
9 1 3x2
34x
1
8 x 1
23 2x2
1
10 1 x2
1,000
10x2 1 103
304 CHAPTER 4 Exponential and Logarithmic Functions 4-14
39. PRESENT VALUE How much money should be
invested now at 7% to obtain $9,000 in 5 years if
interest is compounded:
a. Quarterly
b. Continuously
40. PRESENT VALUE What is the present value of
$10,000 over a 5-year period of time if interest is
compounded continuously at an annual rate of
7%? What is the present value of $20,000 under
the same conditions?
41. DEMAND A manufacturer estimates that when
x units of a particular commodity are produced,
the market price p (dollars per unit) is given by
the demand function
a. What market price corresponds to the production
of x 100 units?
b. How much revenue is obtained when 100 units
of the commodity are produced?
c. How much more (or less) revenue is obtained
when x 100 units are produced than when
x 50 are produced?
42. DEMAND A manufacturer estimates that when
x units of a particular commodity are produced,
the market price p (dollars per unit) is given by
the demand function
a. What market price corresponds to the production
of x 0 units?
b. How much revenue is obtained when 200 units
of the commodity are produced?
c. How much more (or less) revenue is obtained
when x 100 units are produced than when
x 50 are produced?
43. POPULATION GROWTH It is projected that t
years from now, the population of a certain
country will be P(t) 50e0.02t million.
a. What is the current population?
b. What will the population be 30 years from now?
44. POPULATION GROWTH It is estimated
that t years after 2000, the population of a certain
country will be P(t) million people where
a. What was the population in 2000?
b. What will the population be in 2010?
45. DRUG CONCENTRATION The concentration
of drug in a patient's bloodstream t hours after an
P(t) 2 50.018t
p 7 50e x/200
p 300e 0.02x
injection is given by C(t) 3 2 0.75t milligrams
per milliliter (mg/mL).
a. What is the concentration when t 0? After 1
hour?
b. What is the average rate of change of concentration
during the second hour?
46. DRUG CONCENTRATION The concentration
of drug in a patient's bloodstream t hours after an
injection is given by C(t) Ae 0.87t milligrams
per millimeter (mg/ml) for constant A. The
concentration is 4 mg/ml after 1 hour.
a. What is A?
b. What is the initial concentration (t 0)? The
concentration after 2 hours?
c. What is the average rate of change of concentration
during the first two hours?
47. BACTERIAL GROWTH The size of a bacterial
culture grows in such a way that after t minutes,
there are P(t) A 20.001t bacteria present, for
some constant A. After 10 minutes, there are
10,000 bacteria.
a. What is A?
b. How many bacteria are initially present (t 0)?
After 20 minutes? After 1 hour?
c. At what average rate does the bacterial population
change over the second hour?
48. ADVERTISING A marketing manager estimates
that t days after termination of an advertising
campaign, the sales of a new product will be S(t)
units, where
S(t) 4000 e 0.015t
a. How many units are being sold at the time
advertising ends?
b. How many units will be sold 30 days after the
advertising ends? After 60 days?
c. At what average rate do sales change over the
first three months (90 days) after advertising
ends?
49. REAL ESTATE INVESTMENT In 1626, Peter
Minuit traded trinkets worth $24 to a tribe of
Native Americans for land on Manhattan Island.
Assume that in 1990 the same land was worth
$25.2 billion. If the sellers in this transaction had
invested their $24 at 7% annual interest
compounded continuously during the entire
364-year period, who would have gotten the better
end of the deal? By how much?
4-15 SECTION 4.1 EXPONENTIAL FUNCTIONS; CONTINUOUS COMPOUNDING 305
50. GROWTH OF GDP The gross domestic
product (GDP) of a certain country was $500
billion at the beginning of the year 2000 and
increases at the rate of 2.7% per year. (Hint:
Think of this as a compounding problem.)
a. Express the GDP of this country as a function of
the number of years t after 2000.
b. What does this formula predict the GDP of the
country will be at the beginning of the year 2010?
51. POPULATION GROWTH The size of a
bacterial population P(t) grows at the rate of 3.1%
per day. If the initial population is 10,000, what is
the population after 10 days? (Hint: Think of this
as a compounding problem.)
52. SUPPLY A manufacturer will supply
S(x) 300e0.03x 310 units of a particular
commodity when the price is x dollars per unit.
a. How many units will be supplied when the unit
price is $10?
b. How many more units will be supplied when the
unit price is $100 than when it is $80?
53. DRUG CONCENTRATION The concentration
of a certain drug in an organ t minutes after an
injection is given by
C(t) 0.065(1 e 0.025t)
grams per cubic centimeter (g/cm3).
a. What is the initial concentration of drug
(when t 0)?
b. What is the concentration 20 minutes after an
injection? After 1 hour?
c. What is the average rate of change of concentration
during the first minute?
d. What happens to the concentration of the drug
in the long run (as )?
e. Sketch the graph of C(t).
54. DRUG CONCENTRATION The concentration
of a certain drug in an organ t minutes after an
injection is given by
C(t) 0.05 0.04(1 e 0.03t)
grams per cubic centimeter (g/cm3).
a. What is the initial concentration of drug
(when t 0)?
b. What is the concentration 10 minutes after an
injection? After 1 hour?
c. What is the average rate of change of concentration
during the first hour?
d. What happens to the concentration of the drug
in the long run (as )?
e. Sketch the graph of C(t).
t →
t →
306 CHAPTER 4 Exponential and Logarithmic Functions 4-16
In Exercises 55 through 58, find the effective interest
rate re for the given investment.
55. Annual interest rate 6%, compounded quarterly
56. Annual interest rate 8%, compounded daily
(use k 365)
57. Nominal annual rate of 5%, compounded continuously
58. Nominal annual rate of 7.3%, compounded
continuously
59. RANKING INVESTMENTS In terms of
effective interest rate, order the following nominal
rate investments from lowest to highest:
a. 7.9% compounded semiannually
b. 7.8% compounded quarterly
c. 7.7% compounded monthly
d. 7.65% compounded continuously
60. RANKING INVESTMENTS In terms of
effective interest rate, order the following nominal
rate investments from lowest to highest:
a. 4.87% compounded quarterly
b. 4.85% compounded monthly
c. 4.81% compounded daily (365 days)
d. 4.79% compounded continuously
61. EFFECT OF INFLATION Tom buys a rare
stamp for $500. If the annual rate of inflation is
4%, how much should he ask when he sells it in
5 years in order to break even?
62. EFFECT OF INFLATION Suppose during a
10-year period of rapid inflation, it is estimated
that prices inflate at an annual rate of 5% per
year. If an item costs $3 at the beginning of the
period, what would you expect to pay for the
same item 10 years later?
63. PRODUCT RELIABILITY A statistical study
indicates that the fraction of the electric toasters
manufactured by a certain company that are still
in working condition after t years of use is
approximately f (t) e 0.2t.
a. What fraction of the toasters can be expected to
work for at least three years?
b. What fraction of the toasters can be expected to
fail before 1 year of use?
c. What fraction of the toasters can be expected to
fail during the third year of use?
64. LEARNING According to the Ebbinghaus
model, the fraction F(t) of subject matter you will
remember from this course t months after the final
exam can be estimated by the formula
F(t) B (1 B)e kt
where B is the fraction of the material you will
never forget and k is a constant that depends on
the quality of your memory. Suppose you are
tested and it is found that B 0.3 and k 0.2.
What fraction of the material will you remember
one month after the class ends? What fraction will
you remember after one year?
65. POPULATION DENSITY The population
density x miles from the center of a certain city is
D(x) 12e 0.07x thousand people per square mile.
a. What is the population density at the center of
the city?
b. What is the population density 10 miles from
the center of the city?
66. RADIOACTIVE DECAY The amount of a
sample of a radioactive substance remaining
after t years is given by a function of the form
Q(t) Q0e 0.0001t. At the end of 5,000 years,
200 grams of the substance remain. How many
grams were present initially?
67. AQUATIC PLANT LIFE Plant life exists only
in the top 10 meters of a lake or sea, primarily
because the intensity of sunlight decreases
exponentially with depth. Specifically, the
Bouguer-Lambert law says that a beam of light
that strikes the surface of a body of water with
intensity I0 will have intensity I at a depth of x
meters, where I I0e kx with k 0. The constant
k, called the absorption coefficient, depends on
the wavelength of the light and the density of the
water. Suppose a beam of sunlight is only 10% as
intense at a depth of 3 meters as at the surface.
How intense is the beam at a depth of 1 meter?
(Express your answer in terms of I0.)
68. LINGUISTICS Glottochronology is the
methodology used by linguists to determine how
many years have passed since two modern
languages “branched” from a common ancestor.
Experiments suggest that if N words are in
common use at a base time t 0, then the
number N(t) of them still in use with essentially
the same meaning t thousand years later is given
by the so-called fundamental glottochronology
equation*
N(t) N0e 0.217t
*Source: Anthony LoBello and Maurice D. Weir, “Glottochronology:
An Application of Calculus to Linguistics,” UMAP Modules 1982:
Tools for Teaching, Lexington, MA: Consortium for Mathematics and
Its Applications, Inc., 1983.
4-17 SECTION 4.1 EXPONENTIAL FUNCTIONS; CONTINUOUS COMPOUNDING 307
a. Out of a set of 500 basic words used in classical
Latin in 200 B.C., how many would you expect
to be still in use in modern Italian in the year
2010?
b. The research of C. W. Feng and M. Swadesh indicated
that out of a set of 210 words commonly
used in classical Chinese in 950 A.D., 167 were
still in use in modern Mandarin in 1950. Is this
the same number that the fundamental glottochronology
equation would predict? How do
you account for the difference?
c. Read an article on glottochronology and write a
paragraph on its methodology. You may wish to
begin your research by reading the article cited
in this exercise.
69. POPULATION GROWTH It is estimated that t
years after 1990, the population of a certain
country will be P(t) million people where
for certain constants A and B. The population was
100 million in 1992 and 200 million in 2005.
a. Use the given information to find A and B.
b. What was the population in 1990?
c. What will the population be in 2010?
P(t) Ae0.03t Be0.005t
72. MORTGAGE PAYMENTS Suppose a family
figures it can handle monthly mortgage payments
of no more than $1,200. What is the largest
amount of money they can borrow, assuming the
lender is willing to amortize over 30 years at 8%
annual interest compounded monthly?
73. TRUTH IN LENDING You are selling your car
for $6,000. A potential buyer says, “I will pay you
$1,000 now for the car and pay off the rest at
12% interest with monthly payments for 3 years.
Let’s see . . . 12% of the $5,000 is $600 and
$5,600 divided by 36 months is $155.56, but I’ll
pay you $160 per month for the trouble of
carrying the loan. Is it a deal?”
a. If this deal sounds fair to you, I have a perfectly
lovely bridge I think you should consider as
your next purchase. If not, explain why the deal
is fishy and compute a fair monthly payment
(assuming you still plan to amortize the debt of
$5,000 over 3 years at 12%).
b. Read an article on truth in lending and think up
some examples of plausible yet shady deals,
such as the proposed used-car transaction in this
exercise.
74. Two graphs y f (x) and y g(x) are reflections
of one another in the y axis if whenever (a, b) is
a point on one of the graphs, then ( a, b) is a
point on the other, as indicated in the
accompanying figure. Use this criterion to show
that the graphs of
y bxand for b 0, b 1 are
reflections of one another in the y axis.
EXERCISE 74
x
y
y g(x) y f(x)
( a, b) (a, b)
a a
y 1
b x
Amortization of Debt ■ If a loan of A dollars
is amortized over n years at an annual interest rate r
(expressed as a decimal) compounded monthly, the
monthly payments are given by
where i is the monthly interest rate. Use this
formula in Exercises 70 through 73.
r
12
M
Ai
1 (1 i) 12n
70. FINANCE PAYMENTS Determine the monthly
car payment for a new car costing $15,675, if
there is a down payment of $4,000 and the car is
financed over a 5-year period at an annual rate of
6% compounded monthly.
71. MORTGAGE PAYMENTS A home loan is
made for $150,000 at 9% annual interest,
compounded monthly, for 30 years. What is the
monthly mortgage payment on this loan?
75. Complete the following table for .
x 2.2 1.5 0 1.5 2.3
f (x)
76. Program a computer or use a calculator to evaluate
for n 1,000, 2,000, . . . , 50,000.
77. Program a computer or use a calculator to evaluate
for n 1,000, 2,000, . . . , 50,000.
On the basis of these calculations, what can you
1
1
n n
1
1
n n
f(x)
1
2
1
4 x
SECTION 4.2 Logarithmic Functions
Suppose you invest $1,000 at 8% compounded continuously and wish to know how
much time must pass for your investment to double in value to $2,000. According to
the formula derived in Section 4.1, the value of your account after t years will be
1,000e0.08t, so to find the doubling time for your account, you must solve for t in the
equation
1,000e0.08t 2,000
or, by dividing both sides by 1,000,
e0.08t 2
We will answer the question about doubling time in Example 4.2.10. Solving
an exponential equation such as this involves using logarithms, which reverse the
process of exponentiation. Logarithms play an important role in a variety of applications,
such as measuring the capacity of a transmission channel and in the famous
Richter scale for measuring earthquake intensity. In this section, we examine the
basic properties of logarithmic functions and a few applications. We begin with a
definition.
EXAMPLE 4.2.1
Evaluate
a. log10 1,000 b. log2 32 c. log5 1
125
Logarithmic Functions ■ If x is a positive number, then the logarithm of
x to the base b (b 0, b 1), denoted logb x, is the number y such that by x;
that is,
y logb x if and only if by x for x 0
308 CHAPTER 4 Exponential and Logarithmic Functions 4-18
conjecture about the behavior of as n
decreases without bound?
78. Program a computer or use a calculator to
estimate
.
79. Program a computer or use a calculator to
estimate
lim .
n→ 2
5
2n n/3
lim
n→ 1
3
n 2n
1
1
n n
Solution
a. log10 1,000 3 since 103 1,000.
b. log2 32 5 since 25 32.
c. log5 3 since
EXAMPLE 4.2.2
Solve each of the following equations for x:
a. b. log64 16 x c. logx 27 3
Solution
a. By definition, log4 is equivalent to x 41/2 2.
b. log64 16 x means
c. logx 27 3 means
x3 27
x (27)1/3 3
Logarithms were introduced in the 17th century as a computational device, primarily
because they can be used to convert expressions involving products and quotients
into much simpler expressions involving sums and differences. Here are the
rules for logarithms that facilitate such simplification.
x
2
3
4 6x bm bn implies m n
24 (26)x 26x
16 64x
x
1
2
log4 x
1
2
5 3
1
125
.
1
125
4-19 SECTION 4.2 LOGARITHMIC FUNCTIONS 309
Logarithmic Rules ■ Let b be any logarithmic base (b 0, b 1). Then
logb 1 0 and logb b 1
and if u and v are any positive numbers, we have
The equality rule logb u logb v if and only if u v
The product rule logb (uv) logb u logb v
The power rule logb ur r logb u for any real number r
The quotient rule
The inversion rule logb bu u
logb u
v logb u logb v
All these logarithmic rules follow from corresponding exponential rules. For
example,
logb 1 0 since b0 1
logb b 1 since b1 b
To prove the equality rule, let
m logb u and n logb v
so that by definition,
bm u and bn v
Therefore, if
logb u logb v
then m n, so
bm bn
or, equivalently,
u v
as stated in the equality rule for logarithms. Similarly, to prove the product rule for
logarithms, note that
Proofs of the power rule and the quotient rule are left as exercises (see Exercise 78).
Table 4.1 displays the correspondence between basic properties of exponential and
logarithmic functions.
logb (uv) since bm u and bn v
logb (bmbn) product rule for exponentials
logb (bm n) definition of logarithm
logb u logb v m n
equality rule for exponentials
310 CHAPTER 4 Exponential and Logarithmic Functions 4-20
TABLE 4.1 Comparison of Exponential and
Logarithmic Rules
logb (xy) logb x logb y
logb logb x logb y
bxp (bx )p logb xp p logb x
x
y
bx
by bx y
bx by bx y
Exponential Rule Logarithmic Rule
EXAMPLE 4.2.3
Use logarithm rules to rewrite each of the following expressions in terms of log5 2
and log5 3.
a. log5 b. log5 8 c. log5 36 5
3
Solution
a. quotient rule
since log5 5 1
b. log5 8 log5 23 3 log5 2 power rule
c. log5 36 log5 (2232)
log5 22 log5 32 product rule
2 log5 2 2 log5 3 power rule
EXAMPLE 4.2.4
Use logarithmic rules to expand each of the following expressions.
a. log3 (x3y 4) b. c.
Solution
a. log3 (x3y 4) log3 x3 log3 y 4 product rule
3 log3 x ( 4) log3 y power rule
3 log3 x 4 log3 y
b. quotient rule
power rule
c.
product rule
power rule
factor 1 y2
product rule
There is an easy way to obtain the graph of the logarithmic function y logb x from
the graph of the exponential function y bx. The idea is that since y logb x is equivalent
to x b y, the graph of y logb x is the same as the graph of y bx with the
roles of x and y reversed. That is, if (u, v) is a point on the curve y logb x, then
v logbu, or equivalently, u bv, which means that (v, u) is on the graph of y bx.
As illustrated in Figure 4.5a, the points (u, v) and (v, u) are mirror images of one
another in the line y x (see Exercise 79). Thus, the graph of y logb x can be
Graphs of Logarithmic
Functions
3 log7 x
1
2
log7 (1 y)
1
2
log7 (1 y)
3 log7 x
1
2
[log7 (1 y) log7 (1 y)]
3 log7 x
1
2
log7 [(1 y)(1 y)]
3 log7 x
1
2
log7 (1 y2)
log7 x3 log7 (1 y2)1/2
log7 (x3 1 y2) log7 [x3(1 y2)1/2]
5 log2 y 2 log2 x
log2 y5
x2 log2 y5 log2 x2
log2 log7 (x3 1 y2) y5
x2
1 log5 3
log5 5
3 log5 5 log5 3
4-21 SECTION 4.2 LOGARITHMIC FUNCTIONS 311
obtained by simply reflecting the graph of y bx in the line y x, as shown in Figure
4.5b for the case where b 1. To summarize:
FIGURE 4.5 The graph of y logb x for b 1 is obtained by reflecting the graph of
y bx in the line y x.
Figure 4.5b reveals important properties of the logarithmic function f(x) logb x
for the case where b 1. The following box lists these properties along with similar
properties for the case where 0 b 1.
x
y
y = x
y = bx
(0, 1) y = 1ogb x
(1, 0)
(b, 1)
(1, b)
b
(b) The graphs of y = logb x and y = bx
are reflections of one another in the
line y = x.
x
y
v u
y = x
(v, u)
(u, v)
u
v
(a) The point (v, u) is the mirror
image of (u, v) in the line y = x.
Relationship Between the Graphs of y 5 logb x and y 5 bx ■ The
graphs of y logb x and y bx are mirror images of one another in the line
y x. Therefore, the graph of y logb x can be obtained by reflecting the graph
of y bx in the line y x.
312 CHAPTER 4 Exponential and Logarithmic Functions 4-22
Properties of a Logarithmic Function ■ The logarithmic function
has these properties:
1. It is defined and continuous for all x 0.
2. The y axis is a vertical asymptote.
3. The x intercept is (1, 0); there is no y intercept.
4. If b 1, then and
If 0 b 1, then and
5. For all x 0, the function is increasing (graph rising) if b > 1 and
decreasing (graph falling) if 0 < b < 1.
lim
x→
lim logb x
x→0
logb x
lim
x→
lim logb x
x→0
logb x
f (x) logb x (b 0, b
1)
In calculus, the most frequently used logarithmic base is e. In this case, the logarithm
loge x is called the natural logarithm of x and is denoted by ln x (read as “el en x”);
that is, for x 0
y ln x if and only if ey x
The graph of the natural logarithm is shown in Figure 4.6.
The Natural
Logarithm
To evaluate ln a for a particular number a 0, use the LN key on your calculator.
For example, to find ln (2.714), you would press the LN key, and then enter the
number 2.714 to get
ln (2.714) 0.9984 (to four decimal places)
Here is an example illustrating the computation of natural logarithms.
EXAMPLE 4.2.5
Find
a. ln e b. ln 1 c. d. ln 2
Solution
a. According to the definition, ln e is the unique number c such that e ec. Clearly
this number is c 1. Hence, ln e 1.
b. ln 1 is the unique number c such that 1 ec. Since e0 1, it follows that
ln 1 0.
c. ln ln e1/2 is the unique number c such that e1/2 ec; that is, Hence,
ln
d. ln 2 is the unique number c such that 2 ec. The value of this number is not
obvious, and you will have to use your calculator to find that ln 2 0.69315.
EXAMPLE 4.2.6
a. Find ln if ln a 3 and ln b 7.
b. Show that ln ln x.
c. Find x if 2x e3.
1
x
ab
e
1
2
.
c
1
2
e .
ln e
4-23 SECTION 4.2 LOGARITHMIC FUNCTIONS 313
EXPLORE!
Store y ex into Y1, using a
bold graphing style, and y x
into Y2. Set a decimal
window. Since
y In x is equivalent to
ey x, we can graph
y In x as the inverse relation
of ex y using option 8:
Drawlnv of the DRAW (2nd
PRGM) key and writing
DrawInv Y1 on the Home
Screen.
x
y
y = x
y = ex
(0, 1) y = ln x
(1, 0)
(e, 1)
(1, e)
e
FIGURE 4.6 The graph of y ln x.
Solution
a. ln ln (ab)1/2 ln ab (ln a ln b) (3 7) 5
b. ln ln 1 ln x 0 ln x ln x
c. Take the natural logarithm of each side of the equation 2x e3 and solve for
x to get
since ln e 1
Thus,
Two functions f and g with the property that f (g(x)) x and g( f (x)) x, whenever
both composite functions are defined, are said to be inverses of one another.
Such an inverse relationship exists between exponential and logarithmic functions
with base b. For instance, we have
Similarly, if y eln x for x 0, then by definition, ln y ln x, so y x; that is,
eln x y x
This inverse relationship between the natural exponential and logarithmic functions is
especially useful. It is summarized in the following box and used in Example 4.2.7.
EXAMPLE 4.2.7
Solve each of the following equations for x:
a. 3 e20x b. 2 ln x 1
Solution
a. Take the natural logarithm of each side of the equation to get
ln 3 ln e20x or ln 3 20x
Solve for x, using a calculator, to find ln 3:
b. First isolate ln x on the left side of the equation by dividing both sides by 2:
ln x
1
2
x
ln 3
20
1.0986
20
0.0549
The Inverse Relationship Between ex and ln x ■
eln x x for x 0 and ln ex x for all x
ln ex x ln e x(1) x for all x
x
3
ln 2
4.33
x ln 2 ln e3 3 ln e 3
1
x
1
2
1
2
1
2
ab
314 CHAPTER 4 Exponential and Logarithmic Functions 4-24
EXPLORE!
Solve the equation
3 ex ln (x2 1)
by placing the left side of the
equation into Y1 and the right
side into Y2. Use Standard
Window (ZOOM 6) to find the
x values of the intersection
points.
EXPLORE!
Put the function y 10x into
Y1 and graph using a bold
graphing style. Then put y x
into Y2, y ln x into Y3, and
y log x into Y4. What can
you conclude from this series
of graphs?
Then apply the exponential function to both sides of the equation to get
Example 4.2.8 illustrates how to use logarithms to find an exponential function
that fits certain specified information.
EXAMPLE 4.2.8
The population density x miles from the center of a city is given by a function of
the form Q(x) Ae kx. Find this function if it is known that the population density
at the center of the city is 15,000 people per square mile and the density 10 miles
from the center is 9,000 people per square mile.
Solution
For simplicity, express the density in units of 1,000 people per square mile. The fact
that Q(0) 15 tells you that A 15. The fact that Q(10) 9 means that
Taking the logarithm of each side of this equation, you get
Hence the exponential function for the population density is Q(x) 15e 0.051x.
You have already seen how to use the LN key to compute natural logarithms,
and most calculators have a LOG key for computing logarithms to base 10, but what
about logarithms to bases other than e or 10? To be specific, suppose you wish to calculate
the logarithmic number c logb a. You have
definition of the logarithm
power rule
Thus, the logarithm logb a can be computed by finding the ratio of two natural logarithms,
ln a and ln b. To summarize:
c
ln a
ln b
c ln b ln a
ln bc ln a
bc a
c logb a
ln
3
5
10k or k
ln 3/5
10
0.051
9 15e 10k or
3
5
e 10k
eln x e1/2 or x e1/2 e 1.6487
4-25 SECTION 4.2 LOGARITHMIC FUNCTIONS 315
Conversion Formula for Logarithms ■ If a and b are positive numbers
with b 1, then
logb a
ln a
ln b
EXPLORE!
Store f(x) Bx into Y1 and
then g(x) logB x into Y2 as
In(x) In(B). Experiment with
different values of 1 , B , 2,
using the STO key, to
determine for which B values
these two functions intersect,
touch at a point, or are
separated.
EXPLORE!
Refer to Example 4.2.8. Place
0 Q A*e^( K*X) into the
equation solver of your
graphing calculator. Find the
distance from the center of
the city if Q 13,500 people
per square mile. Recall from
the example that the density
at the city center is 15,000
and the density 10 miles from
the city center is 9,000 people
per square mile.
EXAMPLE 4.2.9
Find log5 3.172.
Solution
Using the conversion formula, you find
In the introductory paragraph at the beginning of this section, you were asked how
long it would take for a particular investment to double in value. This question is
answered in Example 4.2.10.
EXAMPLE 4.2.10
If $1,000 is invested at 8% annual interest, compounded continuously, how long will
it take for the investment to double? Would the doubling time change if the principal
were something other than $1,000?
Solution
With a principal of $1,000, the balance after t years is B(t) 1,000e0.08t, so the investment
doubles when B(t) $2,000; that is, when
2,000 1,000e0.08t
Dividing by 1,000 and taking the natural logarithm on each side of the equation, we get
If the principal had been P0 dollars instead of $1,000, the doubling time would
satisfy
which is exactly the same equation we had with P0 $1,000, so once again, the doubling
time is 8.66 years.
The situation illustrated in Example 4.2.10 applies to any quantity Q(t) Q0ekt
with k 0. In particular, since at time t 0, we have Q(0) Q0e0 Q0, the quantity
doubles when
t
ln 2
k
ln 2 kt
2 ekt
2Q0 Q0ekt
2 e0.08t
2P0 P0e0.08t
t
ln 2
0.08
8.66 years
ln 2 0.08t
2 e0.08t
Compounding
Applications
log5 3.172
ln 3.172
ln 5
1.1544
1.6094
0.7172
316 CHAPTER 4 Exponential and Logarithmic Functions 4-26
EXPLORE!
Use the equation solver of your
graphing calculator with the
equation F P*e^(R*T) 0 to
determine how long it will take
for $2,500 to double at 8.5%
compounded continuously.
To summarize:
4-27 SECTION 4.2 LOGARITHMIC FUNCTIONS 317
Doubling Time ■ A quantity Q(t) Q0ekt (k 0) doubles when t d,
where
d
ln 2
k
Determining the time it takes for an investment to double is just one of several
issues an investor may address when comparing various investment opportunities.
Two additional issues are illustrated in Examples 4.2.11 and 4.2.12.
EXAMPLE 4.2.11
How long will it take $5,000 to grow to $7,000 in an investment earning interest at
an annual rate of 6% if the compounding is
a. Quarterly b. Continuous
Solution
a. We use the future value formula B P(1 i)kt with We have B 7,000,
P 5,000, and since r 0.06 and there are k 4 compounding
periods per year. Substituting, we find that
Taking the natural logarithm on each side of this equation, we get
Thus, it will take roughly 5.65 years.
b. With continuous compounding, we use the formula B Pert:
Taking logarithms, we get
So, with continuous compounding, it takes only 5.61 years to reach the investment
objective.
t
ln 1.4
0.06
5.61
0.06t ln 1.4
ln e 0.06t ln 1.4
e 0.06t
7,000
5,000
1.4
7,000 5,000e 0.06t
t
22.6
4
5.65
4t
ln 1.4
ln 1.015
22.6
4t ln 1.015 ln 1.4
ln (1.015)4t ln 1.4
(1.015)4t
7,000
5,000
1.4
7,000 5,000(1.015)4t
i
0.06
4
0.015,
i
r
k
.
318 CHAPTER 4 Exponential and Logarithmic Functions 4-28
EXAMPLE 4.2.12
An investor has $1,500 and wishes it to grow to $2,000 in 5 years. At what annual
rate r compounded continuously must he invest to achieve this goal?
Solution
If the interest rate is r, the future value of $1,500 in 5 years is given by 1,500 er(5).
In order for this to equal $2,000, we must have
Taking natural logarithms on both sides of this equation, we get
so
The annual interest rate is approximately 5.75%.
It has been experimentally determined that a radioactive sample of initial size Q0
grams will decay to Q(t) Q0e kt grams in t years. The positive constant k in this
formula measures the rate of decay, but this rate is usually given by specifying the
amount of time t h required for half a given sample to decay. This time h is
called the half-life of the radioactive substance. Example 4.2.13 shows how halflife
is related to k.
EXAMPLE 4.2.13
Show that a radioactive substance that decays according to the formula Q(t) Q0e kt
has half-life .
Solution
The goal is to find the value of t for which that is,
Divide by Q0 and take the natural logarithm of each side to get
ln
1
2
kh
1
2
Q0 Q0e kh
Q(h)
1
2
Q0;
h
ln 2
k
Radioactive Decay
and Carbon Dating
r
1
5
ln
4
3
0.575
5r ln
4
3
ln e5r ln
4
3
e5r
2,000
1,500
4
3
1,500er(5) 2,000
4-29 SECTION 4.2 LOGARITHMIC FUNCTIONS 319
Thus, the half-life is
as required.
In 1960, W. F. Libby won a Nobel prize for his discovery of carbon dating, a
technique for determining the age of certain fossils and artifacts. Here is an outline
of the technique.*
The carbon dioxide in the air contains the radioactive isotope 14C (carbon-14) as
well as the stable isotope 12C (carbon-12). Living plants absorb carbon dioxide from
the air, which means that the ratio of 14C to 12C in a living plant (or in an animal that
eats plants) is the same as that in the air itself. When a plant or an animal dies, the
absorption of carbon dioxide ceases. The 12C already in the plant or animal remains
the same as at the time of death while the 14C decays, and the ratio of 14C to 12C
decreases exponentially. It is reasonable to assume that the ratio R0 of 14C to 12C in
the atmosphere is the same today as it was in the past, so that the ratio of 14C to 12C
in a sample (e.g., a fossil or an artifact) is given by a function of the form R(t) R0e kt.
The half-life of 14C is 5,730 years. By comparing R(t) to R0, archaeologists can estimate
the age of the sample. Example 4.2.14 illustrates the dating procedure.
EXAMPLE 4.2.14
An archaeologist has found a fossil in which the ratio of 14C to 12C is the ratio
found in the atmosphere. Approximately how old is the fossil?
Solution
The age of the fossil is the value of t for which ; that is, for which
Dividing by R0 and taking logarithms, you find that
ln
1
5
kt
1
5
e kt
1
5
R0 R0e kt
R(t)
1
5
R0
1
5
since ln
1
2
ln 2 1 ln 2
ln 2
k
ln 2
k
h
ln
1
2
k
*For instance, see Raymond J. Cannon, “Exponential Growth and Decay,” UMAP Modules 1977: Tools
for Teaching, Lexington, MA: Consortium for Mathematics and Its Applications, Inc., 1978. More
advanced dating procedures are discussed in Paul J. Campbell, “How Old Is the Earth?” UMAP Modules
1992: Tools for Teaching, Lexington, MA: Consortium for Mathematics and Its Applications, Inc., 1993.
320 CHAPTER 4 Exponential and Logarithmic Functions 4-30
and
In Example 4.2.13, you found that the half-life h satisfies and since 14C has
half-life h 5,730 years, you have
Therefore, the age of the fossil is
That is, the fossil is approximately 13,300 years old.
t
ln 5
k
ln 5
0.000121
13,300
k
ln 2
h
ln 2
5,730
0.000121
h
ln 2
k
,
t
ln
1
5
k
ln 5
k
EXERCISE ■ 4.2
In Exercises 1 and 2, use your calculator to find the
indicated natural logarithms.
1. Find ln 1, ln 2, ln e, ln 5, ln , and ln e2. What
happens if you try to find ln 0 or ln ( 2)? Why?
2. Find ln 7, ln , ln e 3, ln , and ln What
happens if you try to find ln ( 7) or ln ( e)?
In Exercises 3 through 8, evaluate the given expression
using properties of the natural logarithm.
3. ln e3
4.
5. eln 5
6. e2 ln 3
7. e3 ln 2 2 ln 5
8.
In Exercises 9 through 12, use logarithmic rules to
rewrite each expression in terms of log3 2 and log3 5.
9. log3 270
10. log3 (2.5)
ln
e3 e
e1/3
ln e
5
e.
1
e2.1
1
3
1
5
11. log3 100
12.
In Exercises 13 through 20, use logarithmic rules to
simplify each expression.
13. log2 (x4y3)
14. log3 (x5y 2)
15.
16.
17.
18.
19.
20.
In Exercises 21 through 36, solve the given equation
for x.
21. 4x 53
22. log2 x 4
ln 4 x
x3 1 x2
ln (x3e x2)
ln 1
x
1
x2
ln x2(3 x)2/3
x2 x 1
ln (x2 4 x2)
ln 3 x2 x
log3 64
125
23. log3 (2x 1) 2
24. 32x 1 17
25. 2 e0.06x
26.
27. 3 2 5e 4x
28. 2 ln x b
29.
30.
31.
32. ln x 2(ln 3 ln 5)
33. 3x e2
34. ak ekx
35.
36.
37. If log2 x 5, what is ln x?
38. If log10 x 3, what is ln x?
39. If log5 (2x) 7, what is ln x?
40. If log3 (x 5) 2, what is ln x?
41. Find ln if ln a 2 and ln b 3.
42. Find if ln b 6 and ln c 2.
43. COMPOUND INTEREST How quickly will
money double if it is invested at an annual interest
rate of 6% compounded continuously?
44. COMPOUND INTEREST How quickly will
money double if it is invested at an annual interest
rate of 7% compounded continuously?
45. COMPOUND INTEREST Money deposited in
a certain bank doubles every 13 years. The bank
compounds interest continuously. What annual
interest rate does the bank offer?
46. TRIPLING TIME How long will it take for a
quantity of money A0 to triple in value if it is
invested at an annual interest rate r compounded
continuously?
1
a
ln b
c a
1
ab3
5
1 2e x 3
25e0.1x
e0.1x 3
10
ln x
1
3
(ln 16 2 ln 2)
5 3 ln x
1
2
ln x
ln x
t
50
C
1
2
Q0 Q0e 1.2x
4-31 SECTION 4.2 LOGARITHMIC FUNCTIONS 321
47. TRIPLING TIME If an account that earns interest
compounded continuously takes 12 years to
double in value, how long will it take to triple in
value?
48. INVESTMENT The Morenos invest $10,000 in
an account that grows to $12,000 in 5 years. What
is the annual interest rate r if interest is
compounded
a. Quarterly b. Continuously
49. COMPOUND INTEREST A certain bank offers
an interest rate of 6% per year compounded annually.
A competing bank compounds its interest
continuously. What (nominal) interest rate should
the competing bank offer so that the efective
interest rates of the two banks will be equal?
50. CONCENTRATION OF DRUG A drug is
injected into a patient’s bloodstream and t seconds
later, the concentration of the drug is C grams per
cubic centimeter (g/cm3), where
C(t) 0.1(1 3e 0.03t )
a. What is the drug concentration after 10 seconds?
b. How long does it take for the drug concentration
to reach 0.12 g/cm3?
51. CONCENTRATION OF DRUG The
concentration of a drug in a patient’s kidneys at
time t (seconds) is C grams per cubic centimeter
(g/cm3), where
C(t) 0.4(2 0.13e 0.02t )
a. What is the drug concentration after 20 seconds?
After 60 seconds?
b. How long does it take for the drug concentration
to reach 0.75 g/cm3?
52. RADIOACTIVE DECAY The amount of a
certain radioactive substance remaining after
t years is given by a function of the form
Q(t) Q0e 0.003t. Find the half-life of the
substance.
53. RADIOACTIVE DECAY The half-life of radium
is 1,690 years. How long will it take for a 50-gram
sample of radium to be reduced to 5 grams?
54. ADVERTISING The editor at a major
publishing house estimates that if x thousand complimentary
copies are distributed to instructors, the
first-year sales of a new text will be
approximately f (x) 20 12e 0.03x thousand
copies. According to this estimate, approximately
how many complimentary copies should the
editor send out to generate first-year sales of
12,000 copies?
55. GROWTH OF BACTERIA A medical student
studying the growth of bacteria in a certain
culture has compiled these data:
Use these data to find an exponential function of
the form Q(t) Q0ekt expressing the number of
bacteria in the culture as a function of time. How
many bacteria are present after 1 hour?
56. GROSS DOMESTIC PRODUCT An economist
has compiled these data on the gross domestic
product (GDP) of a certain country:
Use these data to predict the GDP in the year
2010 if the GDP is growing:
a. Linearly, so that GDP at b.
b. Exponentially, so that GDP Aekt
57. WORKER EFFICIENCY An efficiency
expert hired by a manufacturing firm has
compiled these data relating workers’ output to
their experience:
Suppose output Q is related to experience t
by a function of the form Q(t) 500 Ae kt.
Find the function of this form that fits the data.
What output is expected from a worker with
1 year’s experience?
58. ARCHAEOLOGY An archaeologist has found a
fossil in which the ratio of 14C to 12C is the
ratio found in the atmosphere. Approximately how
old is the fossil?
59. ARCHAEOLOGY Tests of an artifact
discovered at the Debert site in Nova Scotia show
that 28% of the original 14C is still present.
Approximately how old is the artifact?
60. ARCHAEOLOGY The Dead Sea Scrolls were
written on parchment in about 100 B.C. What
percentage of the original 14C in the parchment
remained when the scrolls were discovered in 1947?
1
3
Experience t (months) 0 6
Output Q (units per hour) 300 410
Year 1990 2000
GDP (in billions) 100 180
Number of minutes 0 20
Number of bacteria 6,000 9,000
322 CHAPTER 4 Exponential and Logarithmic Functions 4-32
61. ART FORGERY A forged painting allegedly
painted by Rembrandt in 1640 is found to have
99.7% of its original 14C. When was it actually
painted? What percentage of the original 14C
should remain if it were legitimate?
62. ARCHAEOLOGY In 1389, Pierre d’Arcis,
the bishop of Troyes, wrote a memo to the pope,
accusing a colleague of passing off “a certain
cloth, cunningly painted,” as the burial shroud
of Jesus Christ. Despite this early testimony of
forgery, the image on the cloth is so compelling
that many people regard it as a sacred relic.
Known as the Shroud of Turin, the cloth was
subjected to carbon dating in 1988. If authentic,
the cloth would have been approximately 1,960
years old at that time.
a. If the Shroud were actually 1,960 years old,
what percentage of the 14C would have
remained?
b. Scientists determined that 92.3% of the
Shroud’s original 14C remained. Based on this
information alone, what was the likely age of
the Shroud in 1988?
63. COOLING Instant coffee is made by adding
boiling water (212°F) to coffee mix. If the air
temperature is 70°F, Newton’s law of cooling
says that after t minutes, the temperature of the
coffee will be given by a function of the form
f(t) 70 Ae kt. After cooling for 2 minutes,
the coffee is still 15°F too hot to drink, but 2
minutes later it is just right. What is this “ideal”
temperature for drinking?
64. DEMOGRAPHICS The world’s population
grows at the rate of approximately 2% per year.
If it is assumed that the population growth is
exponential, then the population t years from
now will be given by a function of the form
P(t) P0e0.02t, where P0 is the current
population. (This formula is derived in Chapter
6.) Assuming that this model of population
growth is correct, how long will it take for the
world’s population to double?
65. SUPPLY AND DEMAND A manufacturer
determines that the supply function for x units
of a particular commodity is S(x) ln (x 2)
and the corresponding demand function is
D(x) 10 ln (x 1).
a. Find the demand price p D(x) when the level
of production is x 10 units.
4-33 SECTION 4.2 LOGARITHMIC FUNCTIONS 323
b. Find the supply price p S(x) when x 100
units.
c. Find the level of production and unit price
that correspond to market equilibrium (where
supply demand).
66. SUPPLY AND DEMAND A manufacturer
determines that the supply function for x units of
a particular commodity is S(x) e0.02x and the
corresponding demand funtion is D(x) 3e 0.03x.
a. Find the demand price p D(x) when the level
of production is x 10 units.
b. Find the supply price p S(x) when x 12
units.
c. Find the level of production and unit price that
correspond to market equilibrium (where
supply demand).
67. SPY STORY Having ransomed his superior in
Exercise 19 of Section 3.5, the spy returns home,
only to learn that his best friend, Sigmund
(“Siggy”) Leiter, has been murdered. The police
say that Siggy’s body was discovered at 1 P.M. on
Thursday, stuffed in a freezer where the
temperature was 10°F. He is also told that the
temperature of the corpse at the time of discovery
was 40°F, and he remembers that t hours after
death a body has temperature
T Ta (98.6 Ta)(0.97)t
where Ta is the air temperature adjacent to the
body. The spy knows the dark deed was done by
either Ernst Stavro Blohardt or André Scélérat. If
Blohardt was in jail until noon on Wednesday
and Scélérat was seen in Las Vegas from noon
Wednesday until Friday, who “iced” Siggy,
and when?
68. SOUND LEVELS A decibel, named for
Alexander Graham Bell, is the smallest increase
of the loudness of sound that is detectable by the
human ear. In physics, it is shown that when two
sounds of intensity I1 and I2 (watts/cm3) occur,
the difference in loudness is D decibels, where
When sound is rated in relation to the threshold of
human hearing (I0 10 12), the level of normal
conversation is about 60 decibels, while a rock
concert may be 50 times as loud (110 decibels).
a. How much more intense is the rock concert than
normal conversation?
D 10 log10 I1
I2
b. The threshold of pain is reached at a sound
level roughly 10 times as loud as a rock concert.
What is the decibel level of the threshold of pain?
69. SEISMOLOGY The magnitude formula for the
Richter scale is
where E is the energy released by the earthquake
(in joules), and E0 104.4 joules is the energy
released by a small reference earthquake used as a
standard of measurement.
a. The 1906 San Francisco earthquake released approximately
5.96 1016 joules of energy. What
was its magnitude on the Richter scale?
b. How much energy was released by the Indian
earthquake of 1993, which measured 6.4 on the
Richter scale?
70. SEISMOLOGY On the Richter scale, the
magnitude R of an earthquake of intensity I is
given by
.
a. Find the intensity of the 1906 San Francisco
earthquake, which measured R 8.3 on the
Richter scale.
b. How much more intense was the San Francisco
earthquake of 1906 than the devastating 1995
earthquake in Kobe, Japan, which measured
R 7.1?
71. LEARNING In an experiment designed to test
short-term memory,* L. R. Peterson and M. J.
Peterson found that the probability p(t) of a
subject recalling a pattern of numbers and letters
t seconds after being given the pattern is
p(t) 0.89[0.01 0.99(0.85)t]
a. What is the probability that the subject can
recall the pattern immediately (t 0)?
b. How much time passes before p(t) drops to 0.5?
c. Sketch the graph of p(t).
72. ENTOMOLOGY It is determined that the
volume of the yolk of a house fly egg shrinks
according to the formula V(t) 5e 1.3t mm3
(cubic millimeters), where t is the number of
R
ln I
ln 10
R
2
3
log10 E
E0
*L. R. Peterson and M. J. Peterson, “Short-Term Retention of
Individual Verbal Items,” Journal of Experimental Psychology,
Vol. 58, 1959, pp. 193–198.
324 CHAPTER 4 Exponential and Logarithmic Functions 4-34
days from the time the egg is produced. The egg
hatches after 4 days.
a. What is the volume of the yolk when the egg
hatches?
b. Sketch the graph of the volume of the yolk over
the time period 0 t 4.
c. Find the half-life of the volume of the yolk; that
is, the time it takes for the volume of the yolk to
shrink to half its original size.
73. RADIOLOGY Radioactive iodine 133I has a halflife
of 20.9 hours. If injected into the bloodstream,
the iodine accumulates in the thyroid gland.
a. After 24 hours, a medical technician scans a
patient’s thyroid gland to determine whether
thyroid function is normal. If the thyroid has
absorbed all of the iodine, what percentage of
the original amount should be detected?
b. A patient returns to the medical clinic 25 hours
after having received an injection of 133I. The
medical technician scans the patient’s thyroid
gland and detects the presence of 41.3% of the
original iodine. How much of the original 133I
remains in the rest of the patient’s body?
74. AIR PRESSURE The air pressure f (s) at a
height of s meters above sea level is given by
f (s) e 0.000125s atmospheres
a. The atmospheric pressure outside an airplane is
0.25 atmosphere. How high is the plane?
b. A mountain climber decides she will wear an
oxygen mask once she has reached an altitude of
7,000 meters. What is the atmospheric pressure
at this altitude?
75. ALLOMETRY Suppose that for the first 6 years
of a moose’s life, its shoulder height H(t) and tipto-
tip antler length A(t) increase with time t (years)
according to the formulas H(t) 125e0.08t and
A(t) 50e0.16t, where H and A are both measured
in centimeters (cm).
a. On the same graph, sketch y H(t) and
y A(t) for the applicable period 0 t 6.
b. Express antler length A as a function of height
H. [Hint: First take logarithms on both sides of
the equation H 125e0.08t to express time t in
terms of H and then substitute into the formula
for A(t).]
76. INVESTMENT An investment firm estimates
that the value of its portfolio after t years is A
million dollars, where
A(t) 300 ln (t 3)
a. What is the value of the account when t 0?
b. How long does it take for the account to double
its initial value?
c. How long does it take before the account is
worth a billion dollars?
77. POPULATION GROWTH A community grows
in such a way that t years from now, its population
is P(t) thousand, where
a. What is the population when t 0?
b. How long does it take for the population to double
its initial value?
c. What is the average rate of growth of the population
over the first 10 years?
78. In each case, use one of the laws of exponents to
prove the indicated law of logarithms.
a. The quotient rule: ln ln u ln v
b. The power rule: ln ur r ln u
79. Show that the reflection of the point (a, b) in the
line y x is (b, a). [Hint: Show that the line
joining (a, b) and (b, a) is perpendicular to y x
and that the distance from (a, b) to y x is the
same as the distance from y x to (b, a).]
EXERCISE 79
80. Sketch the graph of y logb x for 0 b 1 by
reflecting the graph of y bx in the line y x.
Then answer these questions:
a. Is the graph of y logb x rising or falling for
x 0?
b. Is the graph concave upward or concave downward
for x 0?
c. What are the intercepts of the graph? Does
the graph have any horizontal or vertical
asymptotes?
d. What can be said about
lim
x→
logb x and lim
x→0 logb x?
0
x
y
y = x
(b, a)
(a, b)
u
v
P(t) 51 100 ln (t 3)
4-35 SECTION 4.3 DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS 325
SECTION 4.3 Differentiation of Exponential
and Logarithmic Functions
In the examples and exercises examined so far in this chapter we have seen how exponential
functions can be used to model a variety of situations, ranging from compound
interest to population growth and radioactive decay. In order to discuss rates of change
and to determine extreme values in such situations, we need derivative formulas for
exponential functions and their logarithmic counterparts. We obtain the formulas in this
section and examine a few basic applications. Additional exponential and logarithmic
models will be explored in Section 4.4. We begin by showing that the natural exponential
function f(x) ex has the remarkable property of being its own derivative.
The Derivative of ex ■ For every real number
d
dx
(ex) ex
To obtain this formula, let f (x) ex and note that
since eA B eAeB
factor ex out of the limit
It can be shown that
lim
h→0
eh 1
h
1
ex lim
h→0
eh 1
h
lim
h→0
ex eh ex
h
lim
h→0
ex h ex
h
f
(x) lim
h→0
f (x h) f (x)
h
81. Show that if y is a power function of x, so that
y Cxk where C and k are constants, then ln y is
a linear function of ln x. (Hint: Take the logarithm
on both sides of the equation y Cxk.)
82. Use the graphing utility of your calculator to
graph y 10x, y x, and y log10 x on the same
coordinate axes (use [ 5, 5]1 by [ 5, 5]1). How
are these graphs related?
In Exercises 83 through 86 solve for x.
83. x ln (3.42 10 8.1)
84. 3,500e0.31x
e 3.5x
1 257e 1.1x
85. e0.113x 4.72 7.031 x
86. ln (x 3) ln x 5 ln (x2 4)
87. Let a and b be any positive numbers other than 1.
a. Show that (loga b)(logb a) 1.
b. Show that loga x for any x 0.
logb x
logb a
326 CHAPTER 4 Exponential and Logarithmic Functions 4-36
as indicated in Table 4.2. (A formal verification of this limit formula requires methods
beyond the scope of this text.) Thus, we have
as claimed. This derivative formula is used in Example 4.3.1.
EXAMPLE 4.3.1
Differentiate the following functions:
a. b.
Solution
a. Using the product rule, we find
power rule and exponential rule
factor out x and ex
b. To differentiate this function, we use the quotient rule:
The fact that ex is its own derivative means that at each point P(c, ec) on the curve
y ex, the slope is equal to ec, the y coordinate of P (Figure 4.7). This is one of the
most important reasons for using e as the base for exponential functions in calculus.
FIGURE 4.7 At each point P(c, ec) on the graph of y ex, the slope equals ec.
x
y
y = ex
(c, ec)
The slope is ec
factor x2 from the numerator
and combine terms
x2 (3ex xex 6)
(ex 2)2
power rule and
exponential rule
(ex 2) [3x2] x3[ex 0]
(ex 2)2
g
(x)
(ex 2) (x3)
x3(ex 2)
(ex 2)2
xex (x 2)
x2ex (2x)ex
f
(x) x2(ex)
(x2)
ex
g(x)
x3
ex 2
f (x) x2 ex
ex
ex (1)
f
(x) ex lim
h→0
eh 1
h
EXPLORE!
Graph y ex using a modified
decimal window, [ 0.7, 8.7]1
by [ 0.1, 6.1]1. Trace the
curve to any value of x and
determine the value of the
derivative at this point.
Observe how close the
derivative value is to the y
coordinate of the graph.
Repeat this for several values
of x.
TABLE 4.2
h
0.01 1.005017
0.001 1.000500
0.0001 1.000050
0.00001 0.999995
0.0001 0.999950
eh 1
h
By using the chain rule in conjunction with the differentiation formula
we obtain this formula for differentiating general exponential functions.
d
dx
(ex) ex
4-37 SECTION 4.3 DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS 327
The Chain Rule for eu ■ If u(x) is a differentiable function of x, then
d
dx
(eu(x)) eu(x)
du
dx
EXAMPLE 4.3.2
Differentiate the function f (x) .
Solution
Using the chain rule with u x2 1, we find
EXAMPLE 4.3.3
Differentiate the function
Solution
Using the chain rule together with the quotient rule, you get
EXAMPLE 4.3.4
Find the largest and the smallest values of the function f (x) xe2 x on the interval
1 x 1.
Solution
By the product rule
f
(x) x
d
dx
(e2 x) e2 x
d
dx
(x) x(2e2 x) e2 x(1) (2x 1)e2 x
e 3x 3(x2 1) 2x
(x2 1)2 e 3x 3x2 2x 3
(x2 1)2
f
(x)
(x2 1)( 3e 3x ) (2x)e 3x
(x2 1)2
f (x)
e 3x
x2 1
f
(x) ex2 1 d
dx
(x2 1) 2xex2 1
ex2 1
so f
(x) 0 when
Evaluating f (x) at the critical number x and at the endpoints of the interval,
x 1 and x 1, we find that
f ( 1) ( 1)e 2 0.135
e 1 0.184 minimum
f (1) (1)e2 7.389 maximum
Thus, f (x) has its largest value 7.389 at x 1 and its smallest value 0.184 at
x .
Derivatives of Here is the derivative formula for the natural logarithmic function.
Logarithmic Functions
1
2
1
2f
1
2
1
2
x
1
2
2x 1 0 since e2 x 0 for all x
(2x 1)e2x 0
328 CHAPTER 4 Exponential and Logarithmic Functions 4-38
The Derivative of ln x ■ For all
d
dx
(ln x)
1
x
x 0
A proof using the definition of the derivative is outlined in Exercise 88. The
formula can also be obtained as follows using implicit differentiation. Consider the
equation
Differentiating both sides with respect to x, we find that
chain rule
since eln x x
so
divide both sides by x
as claimed. The derivative formula for the natural logarithmic function is used in
Examples 4.3.5 through 4.3.7.
EXAMPLE 4.3.5
Differentiate the function f (x) x ln x.
d
dx
[ln x]
1
x
x
d
dx
[ln x] 1
eln x
d
dx
[ln x] 1
d
dx
[eln x]
d
dx
[x]
eln x x
Solution
Combine the product rule with the formula for the derivative of ln x to get
Using the rules for logarithms can simplify the differentiation of complicated expressions.
In Example 4.3.6, we use the power rule for logarithms before differentiating.
EXAMPLE 4.3.6
Differentiate f (x)
Solution
First, since x2 /3, the power rule for logarithms allows us to write
Then, by the quotient rule, we find
EXAMPLE 4.3.7
Differentiate g(t) (t ln t)3/2.
Solution
The function has the form g(t) u3/ 2, where u t ln t, and by applying the general
power rule, we find
If f (x) ln u(x), where u(x) is a differentiable function of x, then the chain rule
yields the following formula for f
(x).
3
2
(t ln t)1/2 1
1
t
3
2
(t ln t)1/2
d
dt
(t ln t)
g
(t)
d
dt
u3/2
3
2
u1/2
du
dt
2
3 1 4 ln x
x5
2
3 x4 1
x 4x3 ln x
x8
f
(x)
2
3 x4(ln x)
(x4)
ln x
(x4)2
f(x)
ln 3 x2
x4
ln x2/3
x4
2
3
ln x
x4
3
x2
ln 3 x2
x4 .
f
(x) x 1
x ln x 1 ln x
4-39 SECTION 4.3 DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS 329
EXPLORE!
Graph y ln x using a
modified decimal window,
[ 0.7, 8.7]1 by [ 3.1, 3.1]1.
Choose a value of x and construct
the tangent line to the
curve at this x. Observe how
close the slope of the tangent
line is to . Repeat this for
several additional values of x.
1
x
cancel common x3 terms
330 CHAPTER 4 Exponential and Logarithmic Functions 4-40
The Chain Rule for ln u ■ If u(x) is a differentiable function of x, then
for u(x) 0
d
dx
[ln u(x)]
1
u(x)
du
dx
EXAMPLE 4.3.8
Differentiate the function f (x) ln (2x3 1).
Solution
Here, we have f (x) ln u, where u(x) 2x3 1. Thus,
EXAMPLE 4.3.9
Find an equation for the tangent line to the graph of f (x) x ln at the point
where x 1.
Solution
When x 1, we have
so the point of tangency is (1, 1). To find the slope of the tangent line at this point,
we first write
and compute the derivative
Thus, the tangent line passes through the point (1, 1) with slope
so it has the equation
or equivalently,
y
1
2
x
1
2
y 1
x 1
1
2
f
(1) 1
1
2(1)
1
2
f
(x) 1
1
2
1
x 1
1
2x
f (x) x ln x x
1
2
ln x
y f(1) 1 ln ( 1) 1 0 1
x
2(3x2)
2x3 1
6x2
2x3 1
f
(x)
1
u
du
dx
1
2x3 1
d
dx
(2x3 1)
point-slope formula
For instance, to obtain the derivative formula for y logb x, recall that
so we have
You are asked to obtain the derivative formula for y bx in Exercise 93.
EXAMPLE 4.3.10
Differentiate each of the following functions:
a. b.
Solution
Using the chain rule, we find:
a.
b.
Next, we shall examine several applications of calculus involving exponential and logarithmic
functions. In Example 4.3.11, we compute the marginal revenue for a commodity
with logarithmic demand.
EXAMPLE 4.3.11
A manufacturer determines that x units of a particular luxury item will be sold when
the price is p(x) 112 x ln x3 hundred dollars per unit.
Applications
4(x2 log7 x)3 2x
1
x ln 7
g
(x) 4(x2 log7 x)3 [x2 log7 x]
f
(x) [(ln 5)52x 3] [2x 3]
(ln 5)52x 3(2)
f (x) 52x 3 g(x) (x2 log7 x)4
1
x ln b
d
dx
(logb x)
d
dx
ln x
ln b
1
ln b
d
dx
(ln x)
logb x
ln x
ln b
4-41 SECTION 4.3 DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS 331
Derivatives of bx and logb x for Base b > 0, b 1
for all x
and
for all x 0
d
dx
(logb x)
1
x ln b
d
dx
(bx) (ln b)bx
Formulas for differentiating exponential and logarithmic functions with bases
other than e are similar to those obtained for y ex and y ln x. These formulas
are given in the following box.
a. Find the revenue and marginal revenue functions.
b. Use marginal analysis to estimate the revenue obtained from producing the fifth
unit. What is the actual revenue obtained from producing the fifth unit?
Solution
a. The revenue is
R(x) xp(x) x(112 x ln x3) 112x x2(3 ln x)
hundred dollars, and the marginal revenue is
b. The revenue obtained from producing the fifth unit is estimated by the marginal
revenue evaluated at x 4; that is, by
R
(4) 112 3(4) 6(4) ln (4) 66.73
Thus, marginal analysis suggests that the manufacturer will receive approximately
66.73 hundred dollars ($6,673) in revenue by producing the additional unit. The
actual revenue obtained by producing the fifth unit is
R(5) R(4) [112(5) 3(5)2 ln 5] [112(4) 3(4)2 ln 4]
439.29 381.46 57.83
hundred dollars ($5,783).
In Example 4.3.12, we examine exponential demand and use marginal analysis
to determine the price at which the revenue associated with such demand is maximized.
Part of the example deals with the concept of elasticity of demand, which was
introduced in Section 3.4.
EXAMPLE 4.3.12
A manufacturer determines that D( p) 5,000e 0.02p units of a particular commodity
will be demanded (sold) when the price is p dollars per unit.
a. Find the elasticity of demand for this commodity. For what values of p is the
demand elastic, inelastic, and of unit elasticity?
b. If the price is increased by 3% from $40, what is the expected effect on demand?
c. Find the revenue R(p) obtained by selling q D( p) units at p dollars per unit.
For what value of p is the revenue maximized?
Solution
a. According to the formula derived in Section 3.4, the elasticity of demand is given by
p[5,000( 0.02)e 0.02p]
5,000e 0.02p 0.02p
p
5,000e 0.02p [5,000e 0.02p( 0.02)]
E( p)
p
q
dq
dp
R
(x) 112 3 x2 1
x (2x) ln x 112 3x 6x ln x
332 CHAPTER 4 Exponential and Logarithmic Functions 4-42
You find that
so
demand is of unit elasticity when E(p) 0.02p 1; that is, when p 50
demand is elastic when E(p) 0.02p 1; or p 50
demand is inelastic when E(p) 0.02p 1; or p 50
The graph of the demand function, showing levels of elasticity, is displayed in Figure 4.8a.
FIGURE 4.8 Demand and revenue curves for the commodity in Example 4.3.12.
b. When p 40, the demand is
q(40) 5,000e 0.02(40) 2,247 units
and the elasticity of demand is
E( p) 0.02(40) 0.8
Thus, an increase of 1% in price from p $40 will result in a decrease in the quantity
demanded by approximately 0.8%. Consequently, an increase of 3% in price, from
$40 to $41.20, results in a decrease in demand of approximately 2,247[3(0.008)]
54 units, from 2,247 to 2,193 units.
c. The revenue function is
R( p) pq 5,000pe 0.02p
for p 0 (only nonnegative prices have economic meaning), with derivative
R
( p) 5,000( 0.02pe 0.02p e 0.02p)
5,000(1 0.02p)e 0.02p
Since e 0.02p is always positive, R
(p) 0 if and only if
1 0.02p 0 or p 50
To verify that p 50 actually gives the absolute maximum, note that
R ( p) 5,000(0.0004p 0.04)e 0.02p
1
0.02
Elastic
Inelastic
p
q
2,000
0
Unit elasticity
(a) The graph of the demand
function q 5,000e 0.02p
5,000
50
(b) The graph of the revenue
function R 5,000pe 0.02p
p
R
0
100,000 Maximum revenue
50
E(p)
0.02p
0.02p
4-43 SECTION 4.3 DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS 333
EXPLORE!
Based on Example 4.3.12,
store the function y Axe Bx
in Y1 of the equation editor.
For different values of A and B
find the location of the
maximum of y in terms of
A and B. For example, set
A 1 and then vary the value
of B (say, 1, 0.5, and 0.1) to
see where the maximal
functional value occurs. Then
fix B at 0.1 and let A vary
(say, 1, 10, 100). Make a
conjecture about the location
of the maximal y value in
this case.
so
R (50) 5,000[0.0004(50) 0.04]e 0.02(50) 37 0
Thus, the second derivative test tells you that the absolute maximum of R(p) does
indeed occur when p 50 (Figure 4.8b).
Differentiating a function that involves products, quotients, or powers can often be
simplified by first taking the logarithm of the function. This technique, called
logarithmic differentiation, is illustrated in Example 4.3.13.
EXAMPLE 4.3.13
Differentiate the function .
Solution
You could find the derivative using the quotient rule and the chain rule, but the resulting
computation would be somewhat tedious. (Try it!)
A more efficient approach is to take logarithms of both sides of the expression
for f:
Notice that by introducing the logarithm, you eliminate the quotient, the cube root,
and the fourth power.
Now use the chain rule for logarithms to differentiate both sides of this equation
to get
so that
If Q(x) is a differentiable function of x, note that
where the ratio on the right is the relative rate of change of Q(x). That is, the relative
rate of change of a quantity Q(x) can be computed by finding the derivative of
lnQ. This special kind of logarithmic differentiation can be used to simplify the computation
of various growth rates, as illustrated in Example 4.3.14.
d
dx
(ln Q)
Q
(x)
Q(x)
3 x 1
(1 3x)4 1
3
1
x 1
12
1 3x
f
(x) f(x) 1
3
1
x 1
12
1 3x
f
(x)
f(x)
1
3
1
x 1
4 3
1 3x
1
3
1
x 1
12
1 3x
1
3
ln (x 1) 4 ln (1 3x)
ln f (x) ln 3 x 1
(1 3x)4 ln 3 (x 1) ln (1 3x)4
f(x)
3
x 1
(1 3x)4
Logarithmic
Differentiation
334 CHAPTER 4 Exponential and Logarithmic Functions 4-44
EXAMPLE 4.3.14
A country exports three goods, wheat W, steel S, and oil O. Suppose at a particular
time t t0, the revenue (in billions of dollars) derived from each of these
goods is
W(t 0) 4 S(t0) 7 O(t0) 10
and that S is growing at 8%, O is growing at 15%, while W is declining at 3%. At
what relative rate is total export revenue growing at this time?
Solution
Let R W S O. At time t t0, we know that
R(t0) W(t0) S(t0) O(t0) 4 7 10 21
The percentage growth rates can be expressed as
so that
W
(t0) 0.03W(t0) S
(t0) 0.08S(t0) O
(t0) 0.15O(t0)
Thus, at t t0, the relative rate of growth of R is
0.0924
That is, at time t t0, the total revenue obtained from the three exported goods is
increasing at the rate of 9.24%.
0.03(4)
21
0.08(7)
21
0.15(10)
21
0.03W(t0)
R(t0)
0.08S(t0)
R(t0)
0.15O(t0)
R(t0)
0.03W(t0) 0.08S(t0) 0.15O(t0)
R(t0)
0.03W(t0) 0.08S(t0) 0.15O(t0)
W(t0) S(t0) O(t0)
[W
(t0) S
(t0) O
(t0)]
[W(t0) S(t0) O(t0)]
R
(t0)
R(t0)
d(ln R)
dt
d
dt
[ln (W S O)]
t t0
W
(t0)
W(t0)
0.03
S
(t0)
S(t0)
0.08
O
(t0)
O(t0)
0.15
4-45 SECTION 4.3 DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS 335
EXERCISES ■ 4.3
In Exercises 1 through 38, differentiate the given
function.
1. f (x) e5x
2. f (x) 3e4x 1
3. f (x) xex
4.
5. f (x) 30 10e 0.05x
f (x)
ex
x
336 CHAPTER 4 Exponential and Logarithmic Functions 4-46
6.
7. f (x) (x2 3x 5)e6x
8.
9. f (x) (1 3ex)2
10.
11.
12. f (x) e1/x
13. f (x) ln x3
14. f (x) ln 2x
15. f (x) x2 ln x
16.
17.
18.
19.
20. f (x) ex ln x
21. f (x) e 2x x3
22.
23. g(s) (es s 1)(2e s s)
24. F(x) ln (2x3 5x 1)
25.
26. g(u) ln (u2 1)3
27.
28.
29.
30.
31. f (x) ln (e x x)
32. f (s) es ln s
33.
34.
35. f (x)
2x
x
L(x) ln x2 2x 3
x2 2x 1
g(u) ln (u u2 1)
f (x)
ex e x
ex e x
f (t) ln t t
h(x)
e x
x2
f (x)
ex e x
2
h(t)
et t
ln t
f(t) t 2 ln 3 t
f (x) ln x 1
x 1
f (x)
ln x
x
f(x) 3 e2x
f (x) x ln x
f(x) e 3x
f (x) 1 ex
f (x) xe x2
f (x) ex2 2x 1 36.
37. f (x) x log10 x
38.
In Exercises 39 through 46, find the largest and
smallest values of the given function over the
prescribed closed, bounded interval.
39. f(x) e1 x for 0 x 1
40. for 0 x 2
41. for 0 x 2
42. for 0 x 1
43. g(t) t3/2e 2t for 0 t 1
44. for 0 x 1
45.
46. h(s) 2s ln s s2 for 0.5 s 2
In Exercises 47 through 52, find an equation for the
tangent line to y f (x) at the specified point.
47. f (x) xe x; where x 0
48. f (x) (x 1)e 2x; where x 0
49.
50.
51.
52. f (x) x ln x; where x e
In Exercises 53 through 56, find the second derivative
of the given function.
53. f (x) e2x 2e x
54. f (x) ln (2x) x2
55. f (t) t2 ln t
56. g(t) t2e t
In Exercises 57 through 64, use logarithmic
differentiation to find the derivative f
(x).
57. f (x) (2x 3)2(x 5x2)1/2
58. f (x) x2e x(3x 5)3
f (x) x2 ln x; where x 1
f(x)
ln x
x
; where x 1
f (x)
e2x
x2 ; where x 1
f(x)
ln (x 1)
x 1
for 0 x 2
f(x) e 2x e 4x
g(x)
ex
2x 1
f(x) (3x 1)e x
F(x) ex2 2x
f (x)
log2 x
x
f (x) x23x2
59.
60.
61.
62.
63.
64.
MARGINAL ANALYSIS In Exercises 65 through 68,
the demand function q D(p) for a particular
commodity is given in terms of a price p per unit at
which all q units can be sold. In each case:
(a) Find the elasticity of demand and determine
the values of p for which the demand is
elastic, inelastic, and of unit elasticity.
(b) If the price is increased by 2% from $15, what
is the approximate effect on demand?
(c) Find the revenue R(p) obtained by selling q
units at the unit price p. For what value of p
is revenue maximized?
65. D( p) 3,000e 0.04 p
66. D( p) 10,000e 0.025p
67. D( p) 5,000(p 11)e 0.1p
68. D( p)
MARGINAL ANALYSIS In Exercises 69 through 72,
the cost C(x) of producing x units of a particular
commodity is given. In each case:
(a) Find the marginal cost C
(x).
(b) Determine the level of production x for which
the average cost is minimized.
69. C(x) e0.2x
70. C(x) 100e0.01x
71. C(x)
72. C(x)
73. DEPRECIATION A certain industrial machine
depreciates so that its value after t years becomes
Q(t) 20,000e 0.4t dollars.
x2 10xe x
12 x ex/10
A(x)
C(x)
x
10,000e p/10
p 1
f (x) log2
x
f (x) 5x2
f (x)
e 3x 2x 5
(6 5x)4
f (x) (x 1)3(6 x)2 3 2x 1
f (x) 4 2x 1
1 3x
f (x)
(x 2)5
6 3x 5
4-47 SECTION 4.3 DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS 337
a. At what rate is the value of the machine changing
with respect to time after 5 years?
b. At what percentage rate is the value of the
machine changing with respect to time after
t years? Does this percentage rate depend on
t or is it constant?
74. COMPOUND INTEREST Money is deposited
in a bank offering interest at an annual rate of
6% compounded continuously. Find the
percentage rate of change of the balance with
respect to time.
75. POPULATION GROWTH It is projected that t
years from now, the population of a certain
country will become P(t) 50e0.02t million.
a. At what rate will the population be changing
with respect to time 10 years from now?
b. At what percentage rate will the population be
changing with respect to time t years from now?
Does this percentage rate depend on t or is it
constant?
76. COOLING A cool drink is removed from a
refrigerator on a hot summer day and placed in a
room whose temperature is 30° Celsius. According
to Newton’s law of cooling, the temperature of the
drink t minutes later is given by a function of the
form f (t) 30 Ae kt. Show that the rate of
change of the temperature of the drink with
respect to time is proportional to the difference
between the temperature of the room and that of
the drink.
77. MARGINAL ANALYSIS The mathematics
editor at a major publishing house estimates that
if x thousand complimentary copies are distributed
to professors, the first-year sales of a certain new
text will be f (x) 20 15e 0.2x thousand copies.
Currently, the editor is planning to distribute
10,000 complimentary copies.
a. Use marginal analysis to estimate the increase in
first-year sales that will result if 1,000 additional
complimentary copies are distributed.
b. Calculate the actual increase in first-year sales
that will result from the distribution of the additional
1,000 complimentary copies. Is the estimate
in part (a) a good one?
78. ECOLOGY In a model developed by John
Helms,* the water evaporation E(T) for a
ponderosa pine is given by
E(T) 4.6e17.3T/(T 237)
where T (degrees Celsius) is the surrounding air
temperature.
a. What is the rate of evaporation when T 30°C?
b. What is the percentage rate of evaporation? At
what temperature does the percentage rate of
evaporation first drop below 0.5?
79. LEARNING According to the Ebbinghaus model
(recall Exercise 64, Section 4.1), the fraction F(t)
of subject matter you will remember from this
course t months after the final exam can be
estimated by the formula F(t) B (1 B)e kt,
where B is the fraction of the material you will
never forget and k is a constant that depends on the
quality of your memory.
a. Find F
(t) and explain what this derivative
represents.
b. Show that F
(t) is proportional to F B and
interpret this result. [Hint: What does F B
represent in terms of what you remember?]
c. Sketch the graph of F(t) for the case where
B 0.3 and k 0.2.
80. ALCOHOL ABUSE CONTROL Suppose the
percentage of alcohol in the blood t hours after
consumption is given by
C(t) 0.12te t/2
a. At what rate is the blood alcohol level changing
at time t?
b. How much time passes before the blood alcohol
level begins to decrease?
c. Suppose the legal limit for blood alcohol is
0.04%. How much time must pass before the
blood alcohol reaches this level? At what rate is
the blood alcohol level decreasing when it
reaches the legal limit?
81. CONSUMER EXPENDITURE The demand for a
certain commodity is D( p) 3,000 e 0.01p units per
month when the market price is p dollars per unit.
a. At what rate is the consumer expenditure
E(p) pD(p) changing with respect to price p?
b. At what price does consumer expenditure stop
increasing and begin to decrease?
c. At what price does the rate of consumer expenditure
begin to increase? Interpret this result.
82. LEARNING In an experiment to test memory
learning, a subject is confronted by a series of
tasks, and it is found that t minutes after the
experiment begins, the number of tasks
successfully completed is
a. For what values of t is the learning function R(t)
increasing? For what values is it decreasing?
b. When is the rate of change of the learning function
R(t) increasing? When is it decreasing? Interpret
your results.
83. ENDANGERED SPECIES An international
agency determines that the number of individuals
of an endangered species that remain in the wild
t years after a protection policy is instituted may
be modeled by
a. At what rate is the population changing at time
t ? When is the population increasing? When is
it decreasing?
b. When is the rate of change of the population increasing?
When is it decreasing? Interpret your
results.
c. What happens to the population in the “long
run’’ (as t → ∞)?
84. ENDANGERED SPECIES The agency in
Exercise 83 studies a second endangered species
but fails to receive funding to develop a policy of
protection. The population of the species is
modeled by
a. At what rate is the population changing at time
t? When is the population increasing? When is it
decreasing?
N(t)
30 500e 0.3t
1 5e 0.3t
N(t)
600
1 3e 0.02t
R(t)
15(1 e 0.01t )
1 1.5e 0.01t
338 CHAPTER 4 Exponential and Logarithmic Functions 4-48
*John A. Helms, “Environmental Control of Net Photosynthesis in
Naturally Growing Pinus Ponderosa Nets,” Ecology, Winter, 1972, p. 92.
b. When is the rate of change of the population increasing?
When is it decreasing? Interpret your
results.
c. What happens to the population in the “long
run” (as t → ∞)?
85. PLANT GROWTH Two plants grow in such a
way that t days after planting, they are P1(t) and
P2(t) centimeters tall, respectively, where
a. At what rate is the first plant growing at time
t 10 days? Is the rate of growth of the second
plant increasing or decreasing at this time?
b. At what time do the two plants have the same
height? What is this height? Which plant is growing
more rapidly when they have the same height?
86. PER CAPITA GROWTH The national income
I(t) of a particular country is increasing by 2.3%
per year, while the population P(t) of the country
is decreasing at the annual rate of 1.75%. The per
capita income C is defined to be
a. Find the derivative of ln C(t).
b. Use the result of part (a) to determine the percentage
rate of growth of per capita income.
87. REVENUE GROWTH A country exports
electronic components E and textiles T. Suppose
at a particular time t t0, the revenue (in billions
of dollars) derived from each of these goods is
E(t0) 11 and T(t0) 8
and that E is growing at 9%, while T is declining
at 2%. At what relative rate is total export
revenue R E T changing at this time?
88. DERIVATIVE FORMULA FOR In x Prove that
the derivative of f (x) ln xis by
completing these steps.
a. Show that the difference quotient of f (x) can be
expressed as
f (x h) f (x)
h
ln 1
h
x 1/h
f
(x)
1
x
C(t)
I(t)
P(t)
.
P1(t)
21
1 25e 0.3t and P2(t)
20
1 17e 0.6t
b. Let so that Show that the difference
quotient in part (a) can be rewritten as
c. Show that the limit of the expression in part (b)
as is ln [Hint: What is
d. Complete the proof by finding the limit of the difference
quotient in part (a) as . [Hint: How
is this related to the limit you found in part c?]
89. POPULATION GROWTH It is projected that t
years from now, the population of a certain town
will be approximately P(t) thousand people, where
At what rate will the population be changing 10
years from now? At what percentage rate will the
population be changing at that time?
90. A quantity grows so that Q(t) . Find the
percentage rate of change of Q with respect to t.
91. Use a numerical differentiation utility to find f
(c),
where c 0.65 and
Then use a graphing utility to sketch the graph of
f (x) and to draw the tangent line at the point
where x c.
92. Repeat Exercise 91 with the function
f (x) (3.7x2 2x 1)e 3x 2
and c 2.17.
93. For base show that
a. By using the fact that .
b. By using logarithmic differentiation.
bx ex ln b
d
dx
(bx) (ln b)bx
b 0, b
1,
f (x) ln 3 x 1
(1 3x)4
Q0
ekt
t
P(t)
100
1 e 0.2t
h→
lim
n→ 1
1
n n
?]
e1/x
1
x
n→ .
ln 1
1
n n 1/x
n x nh.
x
h
4-49 SECTION 4.3 DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS 339
SECTION 4.4 Applications; Exponential Models
Earlier in this chapter we saw how continuous compounding and radioactive decay can
be modeled using exponential functions. ln this section, we introduce several additional
exponential models from a variety of areas such as business and economics, biology,
psychology, demography, and sociology. We begin with two examples illustrating the
particular issues that arise when sketching exponential and logarithmic graphs.
As when graphing polynomials or rational functions, the key to graphing a function
f (x) involving ex or ln x is to use the derivative f
(x) to find intervals of increase and
decrease and then use the second derivative f (x) to determine concavity.
EXAMPLE 4.4.1
Sketch the graph of f (x) x2 8 ln x.
Solution
The function f (x) is defined only for x 0. Its derivative is
and f
(x) 0 if and only if 2x2 8 or x 2 (since x 0). Testing the sign of f
(x)
for 0 x 2 and for x 2, you obtain the intervals of increase and decrease shown
in the figure.
Notice that the arrow pattern indicates there is a relative minimum at x 2, and since
f (2) 22 8 ln 2 1.5, the minimum point is (2, 1.5).
The second derivative
satisfies f (x) 0 for all x 0, so the graph of f (x) is always concave up and there
are no inflection points.
Checking for asymptotes, you find
and
so the y axis (x 0) is a vertical asymptote, but there is no horizontal asymptote. The
x intercepts are found by using your calculator to solve the equation
x2 8 ln x 0
x 1.2 and x 2.9
lim
x→
lim (x2 8 ln x)
x→0
(x2 8 ln x)
f (x) 2
8
x2
Sign of f
(x)
x
0 2
Min
f
(x) 2x
8
x
2x2 8
x
Curve Sketching
340 CHAPTER 4 Exponential and Logarithmic Functions 4-50
1.2
(2, –1.5)
2.9
x
y
FIGURE 4.9 The graph of
f (x) x2 8 ln x.
EXPLORE!
Refer to Example 4.4.1. Graph
f ( x) in regular style together with
f
( x) in bold, using the modified
decimal window [0, 4.7]1 by
[ 3.1, 3.1]1. Locate the exact
minimum point of the graph
either using the minimum finding
feature of the graphing calculator
applied to f( x) or the rootfinding
feature applied to f
( x).
4-51 SECTION 4.4 APPLICATIONS; EXPONENTIAL MODELS 341
To summarize, the graph falls (down from the vertical asymptote) to the minimum
at (2, 1.5), after which it rises indefinitely, while maintaining a concave up shape.
It passes through the x intercept (1.2, 0) on the way down and through (2.9, 0) as it
rises. The graph is shown in Figure 4.9.
EXAMPLE 4.4.2
Determine where the function
is increasing and decreasing, and where its graph is concave up and concave down.
Find the relative extrema and inflection points and draw the graph.
Solution
The first derivative is
Since is always positive, f
(x) is zero if and only if x 0. Since
, the only critical point is (0, 0.4). By the product rule, the
second derivative is
which is zero if x 1. Since
and
the potential inflection points are (1, 0.24) and ( 1, 0.24).
Plot the critical points and then check the signs of the first and second derivatives
on each of the intervals defined by the x coordinates of these points:
The arrow pattern indicates there is a relative maximum at (0, 0.4), and since the concavity
changes at x 1 (from up to down) and at x 1 (from down to up), both
( 1, 0.24) and (1, 0.24) are inflection points.
Complete the graph, as shown in Figure 4.10, by connecting the key points with
a curve of appropriate shape on each interval. Notice that the graph has no x intercepts
0
Max
Sign of f
(x)
x
Sign of f
(x)
1
Inf
x
1
Inf
f( 1)
e 1/2
2
f (1) 0.24
e 1/2
2
0.24
f (x)
x2
2
e x2/2
1
2
e x2/2
1
2
(x2 1)e x2/2
f(0)
1
2
0.4
e x2/2
f
(x)
x
2
e x2/2
f (x)
1
2
e x2/2
EXPLORE!
Refer to Example 4.4.2. Graph
f (x) in regular style along
with f (x) in bold, using the
modified decimal window
[ 4.7, 4.7]1 by [ 0.5, 0.5]0.1.
Find the x intercepts of f (x)
and explain why they are the x
coordinates of the inflection
points of f (x).
since is always positive and that the graph approaches the x axis as a horizontal
asymptote since e x2/2 approaches zero as x increases without bound.
e x2/2
342 CHAPTER 4 Exponential and Logarithmic Functions 4-52
FIGURE 4.10 The standard normal density function: f (x)
1
2
e x2/2.
f (x)
0.4
0.2
–3 –2 –1 1 2 3
x
NOTE The function whose graph was sketched in Example
4.4.2 is known as the standard normal probability density function and
plays a vital role in probability and statistics. The famous “bell-shape” of the
graph is used by physicists and social scientists to describe the distributions of
IQ scores, measurements on large populations of living organisms, the velocity
of a molecule in a gas, and numerous other important phenomena. ■
Suppose you own an asset whose value increases with time. The longer you hold the
asset, the more it will be worth, but there may come a time when you could do better
by selling the asset and reinvesting the proceeds. Economists determine the optimal
time for selling by maximizing the present value of the asset in relation to the
prevailing rate of interest, compounded continuously. The application of this criterion
is illustrated in Example 4.4.3.
EXAMPLE 4.4.3
Suppose you own a parcel of land whose market price t years from now will be
V(t) 20,000 dollars. If the prevailing interest rate remains constant at 7% compounded
continuously, when will the present value of the market price of the land
be greatest?
Solution
In t years, the market price of the land will be V(t) 20,000 . The present value
of this price is
P(t) V(t)e 0.07t 20,000 e 0.07t 20,000
The goal is to maximize P(t) for t 0. The derivative of P is
P
(t) 20,000e t 0.07t 1
2 t
0.07
e t e t 0.07t
e t
e t
Optimal Holding
Time
f(x)
1
2
e x2/2
EXPLORE!
Store the function
P(x ) 20,000*e^( 0.07x )
from Example 4.4.3 into Y1 of
the equation editor of your
graphing calculator. Find an
appropriate window in order to
view the graph and determine
its maximum value.
x
Thus, P
(t) is undefined when t 0 and P
(t) 0 when
or
Since P
(t) is positive if 0 t 51.02 and negative if t 51.02, it follows
that the graph of P is increasing for 0 t 51.02 and decreasing for t 51.02,
as shown in Figure 4.11. Thus, the present value is maximized in approximately
51 years.
NOTE The optimization criterion used in Example 4.4.3 is not the only way
to determine the optimal holding time. For instance, you may decide to sell the
asset when the percentage rate of growth of the asset’s value just equals the prevailing
rate of interest (7% in the example). Which criterion seems more reasonable
to you? Actually, it does not matter, for the two criteria yield exactly the
same result! A proof of this equivalence is outlined in Exercise 58. ■
A quantity Q(t) is said to experience exponential growthif for k 0
and exponential decay if . Many important quantities in business and
economics, and the physical, social, and biological sciences can be modeled in terms
of exponential growth and decay. The future value of a continuously compounded
investment grows exponentially, as does population in the absence of restrictions. We
discussed the decay of radioactive substances in Section 4.2. Other examples of exponential
decay include present value of a continuously compounded investment, sales
of certain commodities once advertising is discontinued, and the concentration of drug
in a patient’s bloodstream.
Q(t) Q0e kt
Q(t) Q0ekt Exponential Growth
and Decay
t 1
2(0.07) 2
51.02
1
2 t
0.07 0
4-53 SECTION 4.4 APPLICATIONS; EXPONENTIAL MODELS 343
t
(years)
P(t)
Maximum
present value
51.02
FIGURE 4.11 Present value
P(t) 20,000e t 0.07t.
Exponential Growth and Decay ■ A quantity Q(t) grows exponentially
if Q(t) Q0ekt for k 0 and decays exponentially if Q(t) Q0e kt for k 0.
FIGURE 4.12 Exponential change.
Q
Q0
Q = Q0ekt
Q = Q0e kt
Q0
Q
t t
(a) Exponential growth (b) Exponential decay
Typical exponential growth and decay graphs are shown in Figure 4.12. It is customary
to display such graphs only for t 0. Note that the graph of Q(t) Q0ekt
“begins” at Q0 on the vertical axis since
Q(0) Q0ek(0) Q0
Note also that the graph of Q(t) Q0ekt rises sharply since
which means that Q(t) always increases at a rate proportional to its current value, so
the larger the value of Q, the larger the slope. The graph of also begins
at Q0, but falls sharply, approaching the t axis asymptotically. Here is an example of
a business model involving exponential decay.
EXAMPLE 4.4.4
A marketing manager determines that sales of a certain commodity will decline exponentially
once the advertising campaign for the commodity is terminated. It is found
that 21,000 units are being sold at the time of termination and 5 weeks later the sales
level is 19,000 units.
a. Find S0 and k so that S(t) S0e kt gives the sales level t weeks after termination
of advertising. What sales should be expeced 8 weeks after the advertising ends?
b. At what rate are sales changing t weeks after termination of the campaign? What
is the percentage rate of change?
Solution
For simplicity, express sales S in terms of thousands of units.
a. We know that S 21 when t 0 and S 19 when t 5. Substituting t 0
into the formula S(t) S0e kt, we get
S(0) 21 S0e k(0) S0(1) S0
so S0 21 and S(t) 21e kt for all t. Substituting S 19 and t 5, we find
that 19 21e k(5) or
Taking the natural logarithm on each side of this equation, we find that
Thus, for all t 0, we have
S(t) 21e 0.02t
For t 8,
S(8) 21e 0.02(8) 17.9
so the model predicts sales of about 17,900 units 8 weeks after the advertising
campaign ends.
k
1
5
ln 19
21 0.02
5k ln
19
21
ln(e 5k) ln
19
21
e 5k
19
21
Q(t) Q0e kt
Q
(t) Q0k ekt kQ(t)
344 CHAPTER 4 Exponential and Logarithmic Functions 4-54
b. The rate of change of sales is given by the derivative
S
(t) 21[e 0.02t ( 0.02)] 0.02(21)e 0.02t
and the percentage rate of change PR is
That is, sales are declining at the rate of 2% per week.
Notice that the percentage rate of change obtained in Example 4.4.4b is the same
as k, expressed as the percentage. This is no accident, since for any function of the
form Q(t) Q0ert, the percentage rate of change is
For instance, an investment in an account that earns interest at an annual rate of
5% compounded continuously has a future value B Pe0.05t. Thus, the percentage
rate of change of the future value is 100(0.05) 5%. which is exactly what we
would expect.
The graph of a function of the form Q(t) B Ae kt, where A, B, and k are positive
constants, is sometimes called a learning curve. The name arose when psychologists
discovered that for t 0, functions of this form often realistically model the
relationship between the efficiency with which an individual performs a task and the
amount of training time or experience the “learner” has had.
To sketch the graph of Q(t) B Ae kt for t 0, note that
Q
(t) Ae kt( k) Ake kt
and
Q (t) Ake kt( k) Ak2e kt
Since A and k are positive, it follows that Q
(t) 0 and Q (t) 0 for all t, so the
graph of Q(t) is always rising and is always concave down. Furthermore, the vertical
(Q axis) intercept is Q(0) B A, and Q B is a horizontal asymptote since
A graph with these features is sketched in Figure 4.13. The behavior of the learning
curve as reflects the fact that “in the long run,” an individual approaches
his or her learning “capacity,” and additional training time will result in only marginal
improvement in performance efficiency.
EXAMPLE 4.4.5
The rate at which a postal clerk can sort mail is a function of the clerk’s experience.
Suppose the postmaster of a large city estimates that after t months on the job, the
average clerk can sort Q(t) 700 400e 0.5t letters per hour.
t →
lim
t→
Q(t) lim
t→
(B Ae kt) B 0 B
Learning Curves
PR
100Q
(t)
Q(t)
100[Q0ert(r)]
Q0ert 100r
2
PR
100 S
(t)
S(t)
100 [ 0.02(21)e 0.02t]
21e 0.02t
4-55 SECTION 4.4 APPLICATIONS; EXPONENTIAL MODELS 345
y
t
B
Learning
capacity
B – A
0
FIGURE 4.13 A learning
curve y B Ae kt.
a. How many letters can a new employee sort per hour?
b. How many letters can a clerk with 6 months’ experience sort per hour?
c. Approximately how many letters will the average clerk ultimately be able to sort
per hour?
Solution
a. The number of letters a new employee can sort per hour is
Q(0) 700 400e0 300
b. After 6 months, the average clerk can sort
Q(6) 700 400e 0.5(6) 700 400e 3 680 letters per hour
c. As t increases without bound, Q(t) approaches 700. Hence, the average clerk will
ultimately be able to sort approximately 700 letters per hour. The graph of the
function Q(t) is sketched in Figure 4.14.
The graph of a function of the form , where A, B, and k are positive
constants, is called a logistic curve. A typical logistic curve is shown in Figure 4.15.
Notice that it rises steeply like an exponential curve at first, and then turns over and
flattens out, approaching a horizontal asymptote in much the same way as a learning
curve. The asymptotic line represents a “saturation level” for the quantity represented
by the logistic curve and is called the carrying capacity of the quantity.
For instance, in population models, the carrying capacity represents the maximum
number of individuals the environment can support, while in a logistic model for
the spread of an epidemic, the carrying capacity is the total number of individuals
susceptible to the disease, say, those who are unvaccinated or, at worst, the entire
community.
To sketch the graph of Q(t) for t 0, note that
and
Verify these formulas and also verify the fact that Q
(t) 0 for all t, which means
the graph of Q(t) is always rising. The equation Q (t) 0 has exactly one solution;
namely, when
t divide both sides by Bk
ln A
Bk
Bkt ln take logarithms on both sides 1
A ln A
e Bkt add 1 to both sides and divide by A
1
A
1 Ae Bkt 0
Q (t)
AB3k2e Bkt( 1 Ae Bkt )
(1 Ae Bkt )3
Q
(t)
AB2ke Bkt
(1 Ae Bkt )2
B
1 Ae Bkt
Q(t)
B
1 Ae Bkt Logistic Curves
346 CHAPTER 4 Exponential and Logarithmic Functions 4-56
Q(t)
t
700
300
0
FIGURE 4.14 Worker
efficiency
Q(t) 700 400e 0.5t
Q(t)
t
B
B
1 + A
ln A
Bk
0
Carrying
capacity
FIGURE 4.15 A logistic
curve Q(t)
B
1 Ae Bkt
4-57 SECTION 4.4 APPLICATIONS; EXPONENTIAL MODELS 347
As shown in this diagram, there is an inflection point at since the concavity
changes there (from up to down).
The vertical intercept of the logistic curve is
Since Q(t) is defined for all t 0, the logistic curve has no vertical asymptotes, but
y B is a horizontal asymptote since
To summarize, as shown in Figure 4.15, the logistic curve begins at
rises sharply (concave up) until it reaches the inflection point at
and then flattens out as it continues to rise (concave down) toward the horizontal
asymptote y B. Thus, B is the carrying capacity of the quantity Q(t) represented
by the logistic curve, and the inflection point at can be interpreted as
a point of diminishing growth.
Logistic curves often provide accurate models of population growth when environmental
factors such as restricted living space, inadequate food supply, or urban
pollution impose an upper bound on the possible size of the population. Logistic
curves are also often used to describe the dissemination of privileged information or
rumors in a community, where the restriction is the number of individuals susceptible
to receiving such information. Here is an example in which a logistic curve is
used to describe the spread of an infectious disease.
EXAMPLE 4.4.6
Public health records indicate that t weeks after the outbreak of a certain form of
influenza, approximately thousand people had caught the disease.
a. How many people had the disease when it broke out? How many had it 2 weeks
later?
b. At what time does the rate of infection begin to decline?
c. If the trend continues, approximately how many people will eventually contract
the disease?
Q(t)
20
1 19e 1.2t
t
ln A
Bk
t
ln A
Bk
,
Q(0)
B
1 A
,
lim
t→
Q(t) lim
t→
B
1 Ae Bkt
B
1 A(0)
B
Q(0)
B
1 Ae0
B
1 A
x
0
Sign of f
(x)
ln A
Bk
t
ln A
Bk
EXPLORE!
Graph the function in Example
4.4.6 using the window [0, 10]1
by [0, 25]5. Trace out this
function for large values of x.
What do you observe?
Determine a graphic way to
find out when 90% of the
population has caught the
disease.
Solution
a. Since it follows that 1,000 people initially had the disease.
When t 2,
so about 7,343 had contracted the disease by the second week.
b. The rate of infection begins to decline at the inflection point on the graph of Q(t).
By comparing the given formula with the logistic formula
you find that B 20, A 19, and Bk 1.2. Thus, the inflection point occurs when
so the epidemic begins to fade about 2.5 weeks after it starts.
c. Since Q(t) approaches 20 as t increases without bound, it follows that approximately
20,000 people will eventually contract the disease. For reference, the
graph is sketched in Figure 4.16.
An organism such as Pacific salmon or bamboo that breeds only once during its lifetime
is said to be semelparous. Biologists measure the per capita rate of reproduction
of such an organism by the function*
where p(x) is the likelihood of an individual organism surviving to age x and f (x) is
the number of female births to an individual reproducing at age x. The larger the value
of R(x), the more offspring will be produced. Hence, the age at which R(x) is maximized
is regarded as the optimal age for reproduction.
EXAMPLE 4.4.7
Suppose that for a particular semelparous organism, the likelihood of an individual
surviving to age x (years) is given by p(x) e 0.15x and that the number of female
births at age x is f (x) 3x0.85. What is the optimal age for reproduction?
Solution
The per capita rate of increase function for this model is
x 1(ln e 0.15x ln 3 ln x0.85)
x 1( 0.15x ln 3 0.85 ln x)
0.15 (ln 3 0.85 ln x)x 1
power rule for logarithms
product rule for logarithms
R(x)
ln [e 0.15x(3x0.85)]
x
R(x)
ln [p(x) f (x)]
x
Optimal Age
for Reproduction
t
ln A
Bk
ln 19
1.2
2.454
Q(t)
B
1 Ae Bkt,
Q(2)
20
1 19e 1.2(2) 7.343
Q(0)
20
1 19
1,
348 CHAPTER 4 Exponential and Logarithmic Functions 4-58
t
20
1
Q(t)
2.5
FIGURE 4.16 The spread
of an epidemic
Q(t)
20
1 19e 1.2t
*Adapted from Claudia Neuhauser, Calculus for Biology and Medicine, Upper Saddle River, NJ:
Prentice-Hall, 2000, p. 199 (Problem 22).
Differentiating R(x) by the product rule, we find that
R
(x) 0 (ln 3 0.85 ln x)( x 2)
Therefore, R
(x) 0 when
To show that this critical number corresponds to a maximum, we apply the second
derivative test. We find that
(supply the details), and since
R (0.7464) 2.0440 0
it follows that the largest value of R(x) occurs when x 0.7464. Thus, the optimal
age for an individual organism to reproduce is when it is 0.7464 years old (approximately
9 months).
EXERCISES ■ 4.4
R (x)
1.7 ln x 2 ln 3 2.55
x3
x e 0.2925 0.7464
ln x
0.85 ln 3
0.85
0.2925
ln 3 0.85 ln x 0.85 0
ln 3 0.85 ln x 0.85
x2
0.85 1
x x 1
4-59 SECTION 4.4 APPLICATIONS; EXPONENTIAL MODELS 349
2.
3.
x
y
2
x
y
2
1
Each of the curves shown in Exercises 1 through 4 is
the graph of one of the six functions listed here. In
each case, match the given curve to the proper
function.
f1(x) 2 e 2 x f2(x) x ln x5
f6(x) (x 1)e 2x
1.
x
e
y
1
2
f5(x)
ln x5
x
f4(x)
2
1 e f3(x) x
2
1 e x
4.
In Exercises 5 through 20, determine where the given
function is increasing and decreasing and where its
graph is concave upward and concave downward.
Sketch the graph of the function. Show as many key
features as possible (high and low points, points of
inflection, vertical and horizontal asymptotes,
intercepts, cusps, vertical tangents).
5. f (t) 2 et
6. g(x) 3 e x
7. g(x) 2 3ex
8. f (t) 3 2et
9.
10.
11. f (x) xex
12. f (x) xe x
13. f (x) xe2 x
14.
15. f (x) x2e x
16. f (x) ex e x
17.
18. f (x) x ln x (for x 0)
19. f (x) (ln x)2 (for x 0)
20.
21. RETAIL SALES The total number of hamburgers
sold by a national fast-food chain is growing
exponentially. If 4 billion had been sold by 2000
and 12 billion had been sold by 2005, how many
will have been sold by 2010?
f(x)
ln x
x
(for x 0)
f (x)
6
1 e x
f (x) e x2
h(t)
2
1 3e2t
f (x)
2
1 3e 2x
x
y
2
1
350 CHAPTER 4 Exponential and Logarithmic Functions 4-60
22. SALES Once the initial publicity surrounding the
release of a new book is over, sales of the
hardcover edition tend to decrease exponentially.
At the time publicity was discontinued, a certain
book was experiencing sales of 25,000 copies per
month. One month later, sales of the book had
dropped to 10,000 copies per month. What will
the sales be after one more month?
23. POPULATION GROWTH It is estimated that
the population of a certain country grows
exponentially. If the population was 60 million
in 1997 and 90 million in 2002, what will the
population be in 2012?
24. POPULATION GROWTH Based on the estimate
that there are 10 bilion acres of arable land on the
earth and that each acre can produce enough food
to feed 4 people, some demographers believe that
the earth can support a population of no more
than 40 billion people. The population of the earth
was approximately 3 billion in 1960 and 4 billion
in 1975. If the population of the earth were
growing exponentially, when would it reach the
theoretical limit of 40 billion?
25. PRODUCT RELIABILITY A manufacturer of
toys has found that the fraction of its plastic batteryoperated
toy oil tankers that sink in fewer than
t days is approximately f (t) 1 e 0.03t.
a. Sketch this reliability function. What happens to
the graph as t increases without bound?
b. What fraction of the tankers can be expected to
float for at least 10 days?
c. What fraction of the tankers can be expected to
sink between the 15th and 20th days?
26. DEPRECIATION When a certain industrial
machine has become t years old, its resale value
will be V(t) 4,800e t/5 400 dollars.
a. Sketch the graph of V(t). What happens to the
value of the machine as t increases without bound?
b. How much is the machine worth when it is new?
c. How much will the machine be worth after
10 years?
27. COOLING A hot drink is taken outside on a
cold winter day when the air temperature is
5°C. According to a principle of physics called
Newton’s law of cooling, the temperature T (in
degrees Celsius) of the drink t minutes after being
taken outside is given by a function of the form
T(t) 5 Ae kt
where A and k are constants. Suppose the
temperature of the drink is 80°C when it is taken
outside and 20 minutes later is 25°C.
a. Use this information to determine A and k.
b. Sketch the graph of the temperature function
T(t). What happens to the temperature as t increases
indefinitely ?
c. What will the temperature be after 30 minutes?
d. When will the temperature reach 0°C?
28. POPULATION GROWTH It is estimated that
t years from now, the population of a certain
country will be P(t) million.
a. Sketch the graph of P(t).
b. What is the current population?
c. What will be the population 50 years from now?
d. What will happen to the population in the long
run?
29. THE SPREAD OF AN EPIDEMIC Public
health records indicate that t weeks after the
outbreak of a certain form of influenza,
approximately f (t) thousand people
had caught the disease.
a. Sketch the graph of f (t).
b. How many people had the disease initially?
c. How many had caught the disease by the end of
3 weeks?
d. If the trend continues, approximately how many
people in all will contract the disease?
30. RECALL FROM MEMORY Psychologists
believe that when a person is asked to recall a set
of facts, the number of facts recalled after t
minutes is given by a function of the form
Q(t) A(1 e kt), where k is a positive constant
and A is the total number of relevant facts in the
person’s memory.
a. Sketch the graph of Q(t).
b. What happens to the graph as t increases without
bound? Explain this behavior in practical
terms.
31. EFFICIENCY The daily output of a worker who
has been on the job for t weeks is given by a
function of the form Q(t) 40 Ae kt. Initially
the worker could produce 20 units a day, and after
1 week the worker can produce 30 units a day.
How many units will the worker produce per day
after 3 weeks?
2
1 3e 0.8t
20
2 3e 0.06t
(t→ )
4-61 SECTION 4.4 APPLICATIONS; EXPONENTIAL MODELS 351
32. ADVERTISING When professors select texts for
their courses, they usually choose from among the
books already on their shelves. For this reason,
most publishers send complimentary copies of
new texts to professors teaching related courses.
The mathematics editor at a major publishing
house estimates that if x thousand complimentary
copies are distributed, the first-year sales of a
certain new mathematics text will be
approximately f (x) 20 15e 0.2x thousand
copies.
a. Sketch this sales function.
b. How many copies can the editor expect to sell in
the first year if no complimentary copies are
sent out?
c. How many copies can the editor expect to sell in
the first year if 10,000 complimentary copies are
sent out?
d. If the editor’s estimate is correct, what is the
most optimistic projection for the first-year sales
of the text?
33. MARGINAL ANALYSIS The economics editor
at a major publishing house estimates that if x
thousand complimentary copies are distributed to
professors, the first-year sales of a certain new
text will be f (x) 15 20e 0.3x thousand copies.
Currently, the editor is planning to distribute 9,000
complimentary copies.
a. Use marginal analysis to estimate the increase in
first-year sales that will result if 1,000 additional
complimentary copies are distributed.
b. Calculate the actual increase in first-year sales
that will result from the distribution of the additional
1,000 complimentary copies. Is the estimate
in part (a) a good one?
34. LABOR MANAGEMENT A business manager
estimates that when x thousand people are
employed at her firm, the profit will be P(x)
million dollars, where
P(x) ln (4x 1) 3x x2
What level of employment maximizes profit?
What is the maximum profit?
35. CHILDHOOD LEARNING A psychologist
measures a child’s capability to learn and
remember by the function
L(t)
ln (t 1)
t 1
where t is the child’s age in years, for 0 t 5.
Answer these questions about this model.
a. At what age does a child have the greatest learning
capability?
b. At what age is a child’s learning capability increasing
most rapidly?
36. AEROBIC RATE The aerobic rating of a person
x years old is modeled by the function
a. At what age is a person’s aerobic rating largest?
b. At what age is a person’s aerobic rating decreasing
most rapidly?
37. STOCK SPECULATION In a classic paper on
the theory of conflict,* L. F. Richardson claimed
that the proportion p of a population advocating
war or other aggressive action at a time t satisfies
where k and C are positive constants. Speculative
day-trading in the stock market can be regarded
as “aggressive action.” Suppose that initially,
of total daily market volume is attributed to
day-trading and that 4 weeks later, the proportion is
When will the proportion be increasing most
rapidly? What will the proportion be at that time?
38. WORLD POPULATION According to a certain
logistic model, the world’s population (in billions)
t years after 1960 will be approximately
a. If this model is correct, at what rate was the
world’s population increasing with respect to
time in the year 2000? At what percentage rate
was the population increasing at this time?
b. When will the population be growing most
rapidly?
c. Sketch the graph of P(t). What feature occurs
on the graph at the time found in part (b)?
What happens to P(t) “in the long run”?
P(t)
40
1 12e 0.08t
1
100
.
1
200
p(t)
Cekt
1 Cekt
A(x)
110(ln x 2)
x
for x 10
d. Do you think such a population model is reasonable?
Why or why not?
39. MARGINAL ANALYSIS A manufacturer can
produce digital recorders at a cost of $125 apiece
and estimates that if they are sold for x dollars
apiece, consumers will buy approximately
1,000e 0.02 x each week.
a. Express the profit P as a function of x. Sketch
the graph of P(x).
b. At what price should the manufacturer sell the
recorders to maximize profit?
40. THE SPREAD OF AN EPIDEMIC An epidemic
spreads through a community so that t weeks after
its outbreak, the number of people who have been
infected is given by a function of the form
f (t) , where B is the number of residents
in the community who are susceptible to the
disease. If of the susceptible residents were
infected initially and had been infected by the
end of the fourth week, what fraction of the
susceptible residents will have been infected by
the end of the eighth week?
41. OZONE DEPLETION It is known that
fluorocarbons have the effect of depleting ozone
in the upper atmosphere. Suppose it is found that
the amount of original ozone Q0 that remains after
t years is given by
Q Q0e 0.0015t
a. At what percentage rate is the ozone level decreasing
at time t?
b. How many years will it take for 10% of the
ozone to be depleted? At what percentage rate is
the ozone level decreasing at this time?
42. BUSINESS TRAINING A company organizes a
training program in which it is determined that
after t weeks. the average trainee produces
P(t) 50(1 e 0.15t ) units
while a typical new worker without special
training produces
a. How many units does the average trainee produce
during the third week of the training period?
b. Explain how the function F(t) P(t) W(t)
can be used to evaluate the effectiveness of the
training program. Is the program effective if it
W(t) 150t units
1
2
1
5
B
1 Ce kt
352 CHAPTER 4 Exponential and Logarithmic Functions 4-62
*Richardson’s original work appeared in Generalized Foreign
Politics, Monograph Supplement 23 of the British Journal of
Psychology (1939). His work was also featured in the text
Mathematical Models of Arms Control and Disarmament by
T. L. Saaty, New York: John Wiley & Sons, 1968.
lasts just 5 weeks? What if it lasts at least 7
weeks? Explain your reasoning.
43. OPTIMAL HOLDING TIME Suppose you own
a parcel of land whose value t years from now
will be V(t) 8,000 dollars. If the prevailing
interest rate remains constant at 6% per year
compounded continuously, when should you sell
the land to maximize its present value?
44. OPTIMAL HOLDING TIME Suppose your
family owns a rare book whose value t years
from now will be V(t) 200 dollars. If the
prevailing interest rate remains constant at 6% per
year compounded continuously, when will it be
most advantageous for your family to sell the
book and invest the proceeds?
45. OPTIMAL AGE FOR REPRODUCTION
Suppose that for a particular semelparous
organism, the likelihood of an individual surviving
to age x years is p(x) e 0.2x and that the
number of female births to an individual at age x
is f (x) 5x0.9. What is the ideal age for
reproduction for an individual organism of this
species? (See Example 4.4.7.)
46. OPTIMAL AGE FOR REPRODUCTION
Suppose an environmental change affects the
organism in Exercise 45 in such a way that an
individual is only half as likely as before to survive
to age x years. If the number of female births f(x)
remains the same, how is the ideal age for
reproduction affected by this change?
47. RESPONSE TO STIMULATION According to
Hoorweg’s law, when a nerve is stimulated by
discharges from an electrical condenser of
capacitance C, the electric energy required to
elicit a minimal response (a muscle contraction) is
given by
where a, b, and R are positive constants.
a. For what value of C is the energy E(C)
minimized? How do you know you have found
the minimum value? (Your answer will be in
terms of a, b, and R.)
b. Another model for E has the form
E(C) mCek/C
where m and k are constants. What must m and k
be for the two models to have the same minimum
value for the same value of C?
E
C
C aR
b
C 2
e 2t
e t
48. FISHERY MANAGEMENT The manager of a
fishery determines that t weeks after 3,000 fish of
a particular species are hatched, the average weight
of an individual fish will be w(t) 0.8te 0.05t
pounds, for 0 t 20. Moreover, the proportion
of the fish that will still be alive after t weeks is
estimated to be
a. The expected yield E(t) from harvesting after t
weeks is the total weight of the fish that are still
alive. Express E(t) in terms of w(t) and p(t).
b. For what value of t is the expected yield E(t) the
largest? What is the maximum expected yield?
c. Sketch the yield curve y E(t) for 0 t 20.
49. FISHERY MANAGEMENT The manager of a
fishery determines that t days after 1,000 fish of a
particular species are released into a pond, the
average weight of an individual fish will be w(t)
pounds and the proportion of the fish still alive
after t days will be p(t), where
a. The expected yield E(t) from harvesting after t
days is the total weight of the fish that are still
alive. Express E(t) in terms of w(t) and p(t).
b. For what value of t is the expected yield E(t) the
largest? What is the maximum expected yield?
c. Sketch the yield curve y E(t).
50. OPTIMAL HOLDING TIME Suppose you own
a stamp collection that is currently worth $1,200
and whose value increases linearly at the rate of
$200 per year. If the prevailing interest rate
remains constant at 8% per year compounded
continuously, when will it be most advantageous
for you to sell the collection and invest the
proceeds?
51. THE SPREAD OF A RUMOR A traffic
accident was witnessed by 10% of the residents of
a small town, and 25% of the residents had heard
about the accident 2 hours later. Suppose the
number N(t) of residents who had heard about the
accident t hours after it occurred is given by a
function of the form
where B is the population of the town and C and
k are constants.
N(t)
B
1 Ce kt
w(t)
10
1 15e 0.05t and p(t) e 0.01t
p(t)
10
10 t
4-63 SECTION 4.4 APPLICATIONS; EXPONENTIAL MODELS 353
a. Use this information to find C and k.
b. How long does it take for half the residents of
the town to know about the accident?
c. When is the news about the accident spreading
most rapidly? [Hint: See Example 4.4.6(b).]
52. EFFECT OF A TOXIN A medical researcher
determines that t hours from the time a toxin is
introduced to a bacterial colony, the population
will be
P(t) 10,000(7 15e 0.05t te 0.05t)
a. What is the population at the time the toxin is
introduced?
b. When does the maximum bacterial population
occur? What is the maximum population?
c. What eventually happens to the bacterial population
as ?
53. CORPORATE ORGANIZATION A Gompertz
curve is the graph of a function of the general form
N(t)
where A, B, and C are constants. Such curves
are used by psychologists and others to describe
such things as learning and growth within an
organization.*
a. Suppose the personnel director of a large corporation
conducts a study that indicates that after
t years, the corporation will have
employees. How many employees are there
originally (at time t 0)? How many are there
after 5 years? When will there be 300 employees?
How many employees will there be “in the
long run”?
b. Sketch the graph of N(t). Then, on the same
graph sketch the graph of the Gompertz function
How would you describe the relationship between
the two graphs?
54. LEARNING THEORY In a learning model
proposed by C. L. Hull, the habit strength H of an
individual is related to the number r of
reinforcements by the equation
H(r) M(1 e kr )
F(t) 500(0.03) (0.4) t
N(t) 500(0.03)(0.4)t
CABt
t→
a. Sketch the graph of H(r). What happens to H(r)
as ?
b. Show that if the number of reinforcements is
doubled from r to 2r, the habit strength is multiplied
by 1 e kr.
55. CONCENTRATION OF DRUG A function of
the form C(t) Ate kt, where A and k are positive
constants, is called a surge function and is
sometimes used to model the concentration of
drug in a patient’s bloodstream t hours after the
drug is administered. Assume t 0.
a. Find C
(t) and determine the time interval
where the drug concentration is increasing and
where it is decreasing. For what value of t is the
concentration maximized? What is the maximum
concentration? (Your answers will be in
terms of A and k.)
b. Find C (t) and determine time intervals of concavity
for the graph of C(t). Find all points of
inflection and explain what is happening to the
rate of change of drug concentration at the times
that correspond to the inflection points.
c. Sketch the graph of C(t) te kt for k 0.2,
k 0.5, k 1.0, and k 2.0. Describe how the
shape of the graph changes as k increases.
56. CONCENTRATION OF DRUG The
concentration of a certain drug in a patient’s
bloodstream t hours after being administered
orally is assumed to be given by the surge
function C(t) Ate kt, where C is measured in
micrograms of drug per milliliter of blood.
Monitoring devices indicate that a maximum
concentration of 5 occurs 20 minutes after the
drug is administered.
a. Use this information to find A and k.
b. What is the drug concentration in the patient’s
blood after 1 hour?
c. At what time after the maximum concentration
occurs will the concentration be half the
maximum?
d. If you double the time in part (c), will the
resulting concentration of drug be the
maximum? Explain.
57. BUREAUCRATIC GROWTH Parkinson’s law†
states that in any administrative department not
1
4
r→
354 CHAPTER 4 Exponential and Logarithmic Functions 4-64
*A discussion of models of Gompertz curves and other models of
differential growth can be found in an article by Roger V. Jean titled,
“Differential Growth, Huxley’s Allometric Formula, and Sigmoid
Growth,” UMAP Modules 1983: Tools for Teaching, Lexington, MA:
Consortium for Mathematics and Its Applications, Inc., 1984.
†C. N. Parkinson, Parkinson’s Law, Boston, MA: Houghton-Mifflin,
1957.
actively at war, the staff will grow by about 6%
per year, regardless of need.
a. Parkinson applied his law to the size of the
British Colonial Office. He noted that the
Colonial Office had 1,139 staff members in
1947. How many staff members did the law
predict for the year 1954? (The actual number
was 1,661.)
b. Based on Parkinson’s law, how long should it
take for a staff to double in size?
c. Read about Parkinson’s law and write a
paragraph about whether or not you think
it is valid in today’s world. You may wish to
begin your research with the book cited in
this exercise.
58. OPTIMAL HOLDING TIME Let V(t) be the
value of an asset t years from now and assume
that the prevailing annual interest rate remains
fixed at r (expressed as a decimal) compounded
continuously.
a. Show that the present value of the asset
P(t) V(t)e rt has a critical number where
V
(t) V(t)r. (Using economic arguments, it
can be shown that the critical number corresponds
to a maximum.)
b. Explain why the present value of V(t) is
maximized at a value of t where the percentage
rate of change (expressed in decimal form)
equals r.
59. CANCER RESEARCH In Exercise 64, Exercise
set 2.3, you were given a function to model the
production of blood cells. Such models are useful
in the study of leukemia and other so-called
dynamical diseases in which certain physiological
systems begin to behave erratically. An alternative
model* for blood cell production developed by A.
Lasota involves the exponential production
function
p(x) Axse sx/r
where A, s, and r are positive constants and x is
the number of granulocytes (a type of white blood
cell) present.
a. Find the blood cell level x that maximizes the
production function p(x). How do you know the
optimum level is a maximum?
b. If s 1, show that the graph of p(x) has two
inflection points. Sketch the graph. Give a
physical interpretation of the inflection points.
c. Sketch the graph of p(x) in the case where
0 s 1. What is different in this case?
d. Read an article on mathematical methods in the
study of dynamical diseases and write a paragraph
on these methods. A good place to start is
the article referenced in this exercise.
60. MORTALITY RATES An actuary measures the
probability that a person in a certain population
will die at age x by the formula
P(x) 2xe x
where is a constant such that 0 e.
a. Find the maximum value of P(x) in terms of .
b. Sketch the graph of P(x).
c. Read an article about actuarial formulas of this
kind. Write a paragraph on what is represented
by the number .
61. THE SPREAD OF AN EPIDEMIC An
epidemic spreads throughout a community so that
t weeks after its outbreak, the number of residents
who have been infected is given by a function of
the form , where A is the total
number of susceptible residents. Show that the
epidemic is spreading most rapidly when half of
the susceptible residents have been infected.
62. Use the graphing utility of your calculator to sketch
the graph of f (x) x(e x e 2x). Use ZOOM
and TRACE to find the largest value of f (x).
What happens to f (x) as ?
63. MARKET RESEARCH A company is trying to
use television advertising to expose as many
people as possible to a new product in a large
metropolitan area with 2 million possible viewers.
A model for the number of people N (in millions)
who are aware of the product after t days is found
to be
N 2(1 e 0.037t)
Use a graphing utility to graph this function. What
happens as ? (Suggestion: Set the range on
your viewing screen to [0, 200]10 by [0, 3]1.)
t→
x→
f (t)
A
1 Ce kt
4-65 SECTION 4.4 APPLICATIONS; EXPONENTIAL MODELS 355
*W. B. Gearhart and M. Martelli, “A Blood Cell Population Model,
Dynamical Diseases, and Chaos,” UMAP Modules 1990: Tools for
Teaching, Arlington, MA: Consortium for Mathematics and Its
Applications, Inc., 1991.
64. OPTIMAL HOLDING TIME Suppose you win
a parcel of land whose market value t years from
now is estimated to be
dollars. If the prevailing interest rate remains
constant at 7% compounded continuously, when
will it be most advantageous to sell the land?
(Use a graphing utility and ZOOM and TRACE
to make the required determination.)
65. DRUG CONCENTRATION In a classic
paper,* E. Heinz modeled the concentration y(t)
of a drug injected into the body intramuscularly
by the function
where t is the number of hours after the injection
and a, b, and c are positive constants, with b a.
a. When does the maximum concentration occur?
What happens to the concentration “in the long
run”?
b. Sketch the graph of y(t).
c. Write a paragraph on the reliability of the Heinz
model. In particular, is it more reliable when t is
small or large? You may wish to begin your
research with the article cited in this exercise.
66. STRUCTURAL DESIGN When a chain, a
telephone line, or a TV cable is strung between
supports, the curve it forms is called a catenary.
A typical catenary curve is
y 0.125(e4x e 4x)
a. Sketch this catenary curve.
b. Catenary curves are important in architecture.
Read an article on the Gateway Arch to the West
in St. Louis, Missouri, and write a paragraph on
the use of the catenary shape in its design.†
67. ACCOUNTING The double declining balance
formula in accounting is
V(t) V0 1
2
L t
y(t)
c
b a
(e at e bt ) t 0
V(t) 20,000te 0.4t
where V(t) is the value after t years of an article
that originally cost V0 dollars and L is a constant,
called the “useful life” of the article.
a. A refrigerator costs $875 and has a useful life of
8 years. What is its value after 5 years? What is
its annual rate of depreciation?
b. In general, what is the percentage rate of change
of V(t)?
68. SPREAD OF DISEASE In the Think About It
essay at the end of Chapter 3, we examined
several models associated with the AIDS
epidemic. Using a data analysis program, we
obtain the function
C(t) 456 1,234te 0.137t
as a model for the number of cases of AIDS
reported t years after the base year of 1990.
a. According to this model, in what year will the
largest number of cases be reported? What will
the maximum number of reported cases be?
b. When will the number of reported cases be the
same as the number reported in 1990?
69. PROBABILITY DENSITY FUNCTION The
general probability density function has the form
where and are constants, with 0.
a. Show that f (x) has an absolute maximum at
x and inflection points at x and
x .
b. Show that f( c) f( c) for every number
c. What does this tell you about the graph of f (x)?
f (x)
1
2
e (x )2/ 2 2
356 CHAPTER 4 Exponential and Logarithmic Functions 4-66
*E. Heinz, “Probleme bei der Diffusion kleiner Substanzmengen
innerhalb des menschlichen Körpers,” Biochem. Z., Vol. 319, 1949,
pp. 482–492.
†A good place to start is the article by William V. Thayer, “The
St. Louis Arch Problem,” UMAP Modules 1983: Tools for Teaching,
Lexington, MA: Consortium for Mathematics and Its Applications,
Inc., 1984.
4-67 CHAPTER SUMMARY 357
Exponential function y bx (293)
Exponential rules: (296)
bx by if and only if x y
bxby bx y
(bx)y bxy
b0 1
Properties of (295)
It is defined and continuous for all x.
The x axis is a horizontal asymptote.
The y intercept is (0, 1).
If b 1, and
If 0 b 1, and
For all x, it is increasing if b 1 and decreasing if
0 b 1.
The natural exponential base e: (297)
Logarithmic function y logb x (308)
Logarithmic rules: (309)
logb u logb v if and only if u v
logb uv logb u logb v
logb ur r logb u
logb 1 0 and logb b 1
logb bu u
Properties of y logb x (b 0, b
l): (312)
It is defined and continuous for all x 0.
The y axis is a vertical asymptote.
The x intercept is (1, 0).
If b 1, and
If 0 b 1, and
For all x 0, it is increasing if b 1 and
decreasing if 0 b 1.
lim
x→0
logb x
lim
x→
logb x
lim
x→0
lim logb x
x→
logb x
logb u
v logb u logb v
e lim
n→ 1
1
n n
2.71828 . . .
lim
x→
lim bx .
x→
bx 0
lim
x→
lim bx .
x→
bx 0
y bx (b 0, b
1):
bx
by bx y
Graphs of y bx and y logb x (b 1): (312)
Natural exponential and logarithmic functions:
y ex (297)
y ln x (312)
Inversion formulas: (314)
eln x x, for x > 0
ln ex x for all x
Conversion formula for logarithms: (315)
Derivatives of exponential functions: (327)
Derivatives of logarithmic functions: (330)
Logarithmic differentiation (334)
Applications
Compounding interest k times per year at an annual
interest rate r for t years:
Future value of P dollars is (299)
Present value of B dollars is (301)
Effective interest rate is (302)
Continuous compounding at an annual interest rate r
for t years:
Future value of P dollars is B Pert. (299)
Present value of B dollars is P Be–rt. (301)
Effective interest rate is re er 1. (302)
re 1
r
k k
1
P B 1
r
k kt
B P 1
r
k kt
d
dx
(ln x)
1
x
and
d
dx
[ln u(x)]
1
u(x)
du
dx
d
dx
(ex) ex and
d
dx
[eu(x)] eu(x)
du
dx
logb a
ln a
ln b
x
y
y bx y logb x
y x
Important Terms, Symbols, and Formulas
CHAPTER SUMMARY
CHAPTER SUMMARY
358 CHAPTER 4 Exponential and Logarithmic Functions 4-68
Optimal holding time (342)
Exponential growth (343)
Exponential decay (343)
Decay
Q = Q0e kt
Q
Q0
t
Half-life h
ln 2
k
Q Growth
Q0
Q = Q0ekt
t
Doubling time d
ln 2
k
Carbon dating (319)
Learning curve y B Ae kt (345)
Logistic curve (346)
y
t
B
B
1 + A
ln A
Bk
0
Carrying
capacity
y
B
1 Ae Bkt
y
t
B
Learning
capacity
B – A
0
1. Evaluate each of these expressions:
a.
b.
c. log24 log416 1
d.
2. Simplify each of these expressions:
a. (9x4y2)3/2
b. (3x2y4/3) 1/2
c.
d. x0.2y 1.2
x1.5y0.4 5
y
x 3/2 x2/3
y1/6 2
8
27 2/3 16
81 3/2
3
(25)1.5 8
27
(3 2
(92)
(27)2/3
3. Find all real numbers x that satisfy each of these
equations.
a.
b. e1/x 4
c. log4 x2 2
d.
4. In each case, find the derivative . (In some
cases, it may help to use logarithmic
differentiation.)
a.
b. y ln (x3 2x2 3x)
c. y x3 ln x
d. y
e 2x(2x 1
3
1 x2
y
ex
x2 3x
dy
dx
25
1 2e 0.5t 3
42x x2
1
64
Checkup for Chapter 4
CHAPTER SUMMARY
4-69 CHAPTER SUMMARY 359
5. In each of these cases, determine where the given
function is increasing and decreasing and where
its graph is concave upward and concave
downward. Sketch the graph, showing as many
key features as possible (high and low points,
points of inflection, asymptotes, intercepts, cusps,
vertical tangents).
a. y x2e x
b.
c.
d.
6. If you invest $2,000 at 5% compounded
continuously, how much will your account be
worth in 3 years? How long does it take before
your account is worth $3,000?
7. PRESENT VALUE Find the present value of
$8,000 payable 10 years from now if the annual
interest rate is 6.25% and interest is compounded:
a. Semiannually
b. Continuously
8. PRICE ANALYSIS A product is introduced and
t months later, its unit price is p(t) hundred
dollars, where
p
ln (t 1)
t 1
5
y
4
1 e x
y ln ( x x)2
y
ln x
x2
a. For what values of t is the price increasing?
When is it decreasing?
b. When is the price decreasing most rapidly?
c. What happens to the price in the long run
(as )?
9. MAXIMIZING REVENUE It is determined
that q units of a commodity can be sold when the
price is p hundred dollars per unit, where
q(p) 1,000(p 2)e p
a. Verify that the demand function q(p) decreases
as p increases for p 0.
b. For what price p is revenue R pq maximized?
What is the maximum revenue?
10. CARBON DATING An archaeological artifact is
found to have 45% of its original 14C. How old is
the artifact? (Use 5,730 years as the half-life of
14C.)
11. BACTERIAL GROWTH A toxin is introduced
into a bacterial colony, and t hours later, the
population is given by
N(t) 10,000(8 t)e 0.1t
a. What was the population when the toxin was
introduced?
b. When is the population maximized? What is the
maximum population?
c. What happens to the population in the long run
(as t→ )?
t→
In Exercises 1 through 4, sketch the graph of the given
exponential or logarithmic function without using
calculus.
1. f (x) 5x
2. f (x) 2e x
3. f (x) ln x2
4. f (x) log3 x
5. a. Find f (4) if f (x) Ae kx and f (0) 10,
f (1) 25.
b. Find f (3) if f (x) Aekx and f (1) 3,
f (2) 10.
c. Find f (9) if f (x) 30 Ae kx and f (0) 50,
f (3) 40.
d. Find f (10) if and f (0) 3,
f (5) 2.
f(t)
6
1 Ae kt
Review Exercises
6. Evaluate the following expressions without using
tables or a calculator.
a. ln e5
b. eln 2
c. e3 ln 4 ln 2
d. ln 9e2 ln 3e 2
In Exercises 7 through 13, find all real numbers x that
satisfy the given equation.
7. 8 2e0.04x
8. 5 1 4e 6x
9. 4 ln x 8
10. 5x e3
11. log9 (4x 1) 2
12. ln (x 2) 3 ln (x 1)
CHAPTER SUMMARY
360 CHAPTER 4 Exponential and Logarithmic Functions 4-70
13. e2x ex 2 0 [Hint: Let u ex.]
14. e2x 2ex 3 0 [Hint: Let u ex.]
In Exercises 15 through 30, find the derivative . In
some of these problems, you may need to use implicit
differentiation or logarithmic differentiation.
15. y x2e x
16. y 2e3x 5
17. y x ln x2
18.
19. y log3 (x2)
20.
21.
22.
23. y ln (e 2x e x)
24. y (1 e x)4/5
25.
26.
27.
28. xe y ye x 3
29.
30.
In Exercises 31 through 38, determine where the given
function is increasing and decreasing and where its
graph is concave upward and concave downward. Sketch
the graph, showing as many key features as possible
(high and low points, points of inflection, asymptotes,
intercepts, cusps, vertical tangents).
31. f (x) ex e x
32. f (x) xe 2x
33. f (t) t e t
y
e 2x(2 x3)3/2
1 x2
y
(x2 e2x)3e 2x
(1 x x2)2/3
yex x2
x y
y ln e3x
1 x
y
e x
x ln x
y
e3x
e3x 2
y
e x ex
1 e 2x
y
x
ln 2x
y ln x2 4x 1
dy
dx
34.
35. F(u) u2 2 ln (u 2)
36.
37. G(x) ln (e 2x e x)
38. f (u) e2u e u
In Exercises 39 through 42, find the largest and smallest
values of the given function over the prescribed closed,
bounded interval.
39. f (x) ln (4x x2) for 1 x 3
40.
41. h(t) (e t et )5 for 1 t 1
42. for 1 t 2
In Exercises 43 through 46, find an equation for the tangent
line to the given curve at the specified point.
43. y x ln x2 where x 1
44. y (x2 x)e x where x 0
45. y x3e2 x where x 2
46. y (x ln x)3 where x 1
47. Find f (9) if f (x) ekx and f (3) 2.
48. Find f (8) if f (x) A2kx, f (0) 20, and f (2) 40.
49. COMPOUND INTEREST A sum of money is
invested at a certain fixed interest rate, and the
interest is compounded quarterly. After 15 years,
the money has doubled. How will the balance at
the end of 30 years compare with the initial
investment?
50. COMPOUND INTEREST A bank pays 5%
interest compounded quarterly, and a savings
institution pays 4.9% interest compunded
continuously. Over a 1-year period, which account
pays more interest? What about a 5-year period?
51. RADIOACTIVE DECAY A radioactive
substance decays exponentially. If 500 grams of
the substance were present initially and 400 grams
are present 50 years later, how many grams will
be present after 200 years?
g(t)
ln ( t)
t 2
f (x)
e x 2
x2 for 5 x 1
g(t)
ln (t 1)
t 1
f(x)
4
1 e x
CHAPTER SUMMARY
4-71 CHAPTER SUMMARY 361
52. COMPOUND INTEREST A sum of money is
invested at a certain fixed interest rate, and the
interest is compounded continuously. After 10
years, the money has doubled. How will the
balance at the end of 20 years compare with the
initial investment?
53. GROWTH OF BACTERIA The following data
were compiled by a researcher during the first 10
minutes of an experiment designed to study the
growth of bacteria:
Number of minutes 0 10
Number of bacteria 5,000 8,000
Assuming that the number of bacteria grows
exponentially, how many bacteria will be present
after 30 minutes?
54. RADIOACTIVE DECAY The following data
were compiled by a researcher during an
experiment designed to study the decay of a
radioactive substance:
Number of hours 0 5
Grams of substance 1,000 700
Assuming that the sample of radioactive substance
decays exponentially, how much is left after 20
hours?
55. SALES FROM ADVERTISING It is estimated
that if x thousand dollars are spent on advertising,
approximately Q(x) 50 40e 0.1x thousand
units of a certain commodity will be sold.
a. Sketch the sales curve for x 0.
b. How many units will be sold if no money is
spent on advertising?
c. How many units will be sold if $8,000 is spent
on advertising?
d. How much should be spent on advertising to
generate sales of 35,000 units?
e. According to this model, what is the most optimistic
sales projection?
56. WORKER PRODUCTION An employer
determines that the daily output of a worker on
the job for t weeks is Q(t) 120 Ae kt units.
Initially, the worker can produce 30 units per day,
and after 8 weeks, can produce 80 units per day.
How many units can the worker produce per day
after 4 weeks?
57. COMPOUND INTEREST How quickly will
$2,000 grow to $5,000 when invested at an annual
interest rate of 8% if interest is compounded:
a. Quarterly
b. Continuously
58. COMPOUND INTEREST How much should
you invest now at an annual interest rate of 6.25%
so that your balance 10 years from now will be
$2,000 if interest is compounded:
a. Monthly
b. Continuously
59. DEBT REPAYMENT You have a debt of
$10,000, which is scheduled to be repaid at the
end of 10 years. If you want to repay your debt
now, how much should your creditor demand if
the prevailing interest rate is:
a. 7% compounded monthly
b. 6% compounded continuously
60. COMPOUND INTEREST A bank compounds
interest continuously. What (nominal) interest rate
does it offer if $1,000 grows to $2,054.44 in
12 years?
61. EFFECTIVE RATE OF INTEREST Which
investment has the greater effective interest rate:
8.25% per year compounded quarterly or 8.20%
per year compounded continuously?
62. DEPRECIATION The value of a certain
industrial machine decreases exponentially. If the
machine was originally worth $50,000 and was
worth $20,000 five years later, how much will it
be worth when it is 10 years old?
63. POPULATION GROWTH It is estimated that
t years from now the population of a certain country
will be P million people, where
a. Sketch the graph of P(t).
b. What is the current population?
c. What will be the population in 20 years?
d. What happens to the population in “the long
run”?
64. BACTERIAL GROWTH The number of bacteria
in a certain culture grows exponentially. If 5,000
bacteria were initially present and 8,000 were
P(t)
30
1 2e 0.05t
CHAPTER SUMMARY
362 CHAPTER 4 Exponential and Logarithmic Functions 4-72
present 10 minutes later, how long will it take for
the number of bacteria to double?
65. AIR POLLUTION An environmental study of a
certain suburban community suggests that t years
from now, the average level of carbon monoxide
in the air will be Q(t) 4e0.03t parts per million.
a. At what rate will the carbon monoxide level be
changing with respect to time 2 years from now?
b. At what percentage rate will the carbon monoxide
level be changing with respect to time t years
from now? Does this percentage rate of change
depend on t or is it constant?
66. PROFIT A manufacturer of digital cameras
estimates that when cameras are sold for x dollars
apiece, consumers will buy 8000e 0.02x cameras
each week. He also determines that profit is
maximized when the selling price x is 1.4 times
the cost of producing each unit. What price
maximizes weekly profit? How many units are
sold each week at this optimal price?
67. OPTIMAL HOLDING TIME Suppose you own
an asset whose value t years from now will be
dollars. If the prevailing interest
rate remains constant at 5% per year compounded
continuously, when will it be most advantageous
to sell the collection and invest the proceeds?
68. RULE OF 70 Investors are often interested in
knowing how long it takes for a particular
investment to double. A simple means for making
this determination is the “rule of 70,” which says:
The doubling time of an investment with an
annual interest rate r (expressed as a decimal)
compounded continuously is given by .
a. For interest rate r, use the formula B Pert to
find the doubling time for r 4, 6, 9, 10, and
12. In each case, compare the value with the
value obtained from the rule of 70.
b. Some people prefer a “rule of 72” and others
use a “rule of 69.” Test these alternative rules as
in part (a) and write a paragraph on which rule
you would prefer to use.
69. RADIOACTIVE DECAY A radioactive
substance decays exponentially with half-life .
Suppose the amount of the substance initially
present (when t 0) is Q0.
d
70
r
V(t) 2,000e 2t
a. Show that the amount of the substance that remains
after t years will be Q(t) Q0e (ln 2/ )t.
b. Find a number k so that the amount in part (a)
can be expressed as Q(t) Q0(0.5)kt.
70. ANIMAL DEMOGRAPHY A naturalist at an
animal sanctuary has determined that the function
provides a good measure of the number of
animals in the sanctuary that are x years old.
Sketch the graph of f (x) for x 0 and find the
most “likely” age among the animals; that is, the
age for which f (x) is largest.
71. CARBON DATING “Ötzi the Iceman” is the
name given a neolithic corpse found frozen in an
Alpine glacier in 1991. He was originally thought
to be from the Bronze Age because of the hatchet
he was carrying. However, the hatchet proved to
be made of copper rather than bronze. Read an
article on the Bronze Age and determine the least
age of the Ice Man assuming that he dates before
the Bronze Age. What is the largest percentage of
14C that can remain in a sample taken from his
body?
72. FICK’S LAW Fick’s law* says that f (t)
C(1 e kt), where f (t) is the concentration of
solute inside a cell at time t, C is the (constant)
concentration of solute surrounding the cell, and k
is a positive constant. Suppose that for a particular
cell, the concentration on the inside of the cell
after 2 hours is 0.8% of the concentration outside
the cell.
a. What is k?
b. What is the percentage rate of change of f (t) at
time t?
c. Write a paragraph on the role played by Fick’s
law in ecology.
73. COOLING A child falls into a lake where the
water temperature is 3°C. Her body temperature
after t minutes in the water is T(t) 35e 0.32t.
She will lose consciousness when her body
temperature reaches 27°C. How long do rescuers
f(x)
4e (ln x)2
x
*Fick’s law plays an important role in ecology. For instance, see
M. D. LaGrega, P. L. Buckingham, and J. C. Evans, Hazardous
Waste Management, New York: McGraw-Hill, 1994, pp. 95, 464, and
especially p. 813, where the authors discuss contaminant transport
through landfill.
CHAPTER SUMMARY
4-73 CHAPTER SUMMARY 363
have to save her? How fast is her body
temperature dropping at the time it reaches 27°C?
74. FORENSIC SCIENCE The temperature T of the
body of a murder victim found in a room where
the air temperature is 20°C is given by
T(t) 20 17e 0.07t °C
where t is the number of hours after the victim’s
death.
a. Graph the body temperature T(t) for t 0. What
is the horizontal asymptote of this graph and
what does it represent?
b. What is the temperature of the victim’s body
after 10 hours? How long does it take for the
body’s temperature to reach 25°C?
c. Abel Baker is a clerk in the firm of Dewey,
Cheatum, and Howe. He comes to work early
one morning and finds the corpse of his boss,
Will Cheatum, draped across his desk. He calls
the police, and at 8 A.M., they determine that the
temperature of the corpse is 33°C. Since the last
item entered on the victim’s notepad was, “Fire
that idiot, Baker,” Abel is considered the prime
suspect. Actually, Abel is bright enough to have
been reading this text in his spare time. He
glances at the thermostat to confirm that the
room temperature is 20°C. For what time will he
need an alibi in order to establish his innocence?
75. CONCENTRATION OF DRUG Suppose that
t hours after an antibiotic is administered orally, its
concentration in the patient’s bloodstream is given
by a surge function of the form C(t) Ate kt,
where A and k are positive constants and C is
measured in micrograms per milliliter of blood.
Blood samples are taken periodically, and it is
determined that the maximum concentration of
drug occurs 2 hours after it is administered and is
10 micrograms per milliliter.
a. Use this information to determine A and k.
b. A new dose will be administered when the concentration
falls to 1 microgram per milliliter.
When does this occur?
76. CHEMICAL REACTION RATE The effect of
temperature on the reaction rate of a chemical
reaction is given by the Arrhenius equation
where k is the rate constant, T (in kelvin) is the
temperature, and R is the gas constant. The
k Ae E0/RT
quantities A and E0 are fixed once the reaction is
specified. Let k1 and k2 be the reaction rate
constants associated with temperatures T1 and T2.
Find an expression for ln in terms of E0, R,
T1, and T2.
77. POPULATION GROWTH According to a logistic
model based on the assumption that the earth can
support no more than 40 billion people, the world’s
population (in billions) t years after 1960 is
given by a function of the form
where C and k are positive constants. Find the function
of this form that is consistent with the fact that
the world’s population was approximately 3 billion in
1960 and 4 billion in 1975. What does your model
predict for the population in the year 2000? Check
the accuracy of the model by consulting an almanac.
78. THE SPREAD OF AN EPIDEMIC Public
health records indicate that t weeks after the
outbreak of a certain form of influenza,
approximately
thousand people had caught the disease. At what
rate was the disease spreading at the end of the
second week? At what time is the disease
spreading most rapidly?
79. ACIDITY (pH) OF A SOLUTION The acidity
of a solution is measured by its pH value, which
is defined by pH log10 [H3O ], where [H3O ]
is the hydronium ion concentration (moles/liter)
of the solution. On average, milk has a pH value
that is three times the pH value of a lime, which
in turn has half the pH value of an orange. If the
average pH of an orange is 3.2, what is the
average hydronium ion concentration of a lime?
80. CARBON DATING A Cro-Magnon cave
painting at Lascaux, France, is approximately
15,000 years old. Approximately what ratio of 14C
to 12C would you expect to find in a fossil found
in the same cave as the painting?
81. MORTALITY RATES It is sometimes useful for
actuaries to be able to project mortality rates
within a given population. A formula sometimes
used for computing the mortality rate D(t) for
women in the age group 25–29 is
D(t) (D0 0.00046)e 0.162t 0.00046
Q(t)
80
4 76e 1.2t
P(t)
40
1 Ce kt
k1
k2
CHAPTER SUMMARY
364 CHAPTER 4 Exponential and Logarithmic Functions 4-74
where t is the number of years after a fixed base
year and D0 is the mortality rate when t 0.
a. Suppose the initial mortality rate of a particular
group is 0.008 (8 deaths per 1,000 women).
What is the mortality rate of this group 10 years
later? What is the rate 25 years later?
b. Sketch the graph of the mortality function D(t)
for the group in part (a) for 0 t 25.
82. GROSS DOMESTIC PRODUCT The gross
domestic product (GDP) of a certain country was
100 billion dollars in 1990 and 165 billion dollars
in 2000. Assuming that the GDP is growing
exponentially, what will it be in the year 2010?
83. ARCHAEOLOGY “Lucy,” the famous prehuman
whose skeleton was discovered in Africa, has been
found to be approximately 3.8 million years old.
a. Approximately what percentage of original 14C
would you expect to find if you tried to apply carbon
dating to Lucy? Why would this be a problem
if you were actually trying to “date” Lucy?
b. In practice, carbon dating works well only for
relatively “recent” samples—those that are no
more than approximately 50,000 years old. For
older samples, such as Lucy, variations on
carbon dating have been developed, such as
potassium-argon and rubidium-strontium dating.
Read an article on alternative dating methods
and write a paragraph on how they are used.*
84. RADIOLOGY The radioactive isotope
gallium-67 (67Ga), used in the diagnosis of
malignant tumors, has a half-life of 46.5 hours. If
we start with 100 milligrams of the isotope, how
many milligrams will be left after 24 hours? When
will there be only 25 milligrams left? Answer
these questions by first using a graphing utility to
graph an appropriate exponential function and then
using the TRACE and ZOOM features.
85. A population model developed by the U.S. Census
Bureau uses the formula
to estimate the population of the United States (in
millions) for every tenth year from the base year
P(t)
202.31
1 e3.938 0.314t
1790. Thus, for instance, t 0 corresponds to
1790, t 1 to 1800, t 10 to 1890, and so on.
The model excludes Alaska and Hawaii.
a. Use this formula to compute the population of
the United States for the years 1790, 1800,
1830, 1860, 1880, 1900, 1920, 1940, 1960,
1980, 1990, and 2000.
b. Sketch the graph of P(t). When does this model
predict that the population of the United States
will be increasing most rapidly?
c. Use an almanac or some other source to find the
actual population figures for the years listed in
part (a). Does the given population model seem
to be accurate? Write a paragraph describing
some possible reasons for any major differences
between the predicted population figures and the
actual census figures.
86. Use a graphing utility to graph y 2 x, y 3 x,
y 5 x, and y (0.5) x on the same set of axes.
How does a change in base affect the graph of the
exponential function? (Suggestion: Use the
graphing window [ 3, 3]1 by [ 3, 3]1.)
87. Use a graphing utility to draw the graphs of
y , y , and y 3 x on the same set
of axes. How do these graphs differ? (Suggestion:
Use the graphing window [ 3, 3]1 by [ 3, 3]1.)
88. Use a graphing utility to draw the graphs of y 3x
and y 4 ln on the same axes. Then use
TRACE and ZOOM to find all points of
intersection of the two graphs.
89. Solve this equation with three decimal place
accuracy:
log5 (x 5) log2 x 2 log10 (x2 2x)
90. Use a graphing utility to draw the graphs of
y ln (1 x2) and y
on the same axes. Do these graphs intersect?
91. Make a table for the quantities and
, with n 8, 9, 12, 20, 25, 31, 37,
38, 43, 50, 100, and 1,000. Which of the two
quantities seems to be larger? Do you think this
inequality holds for all n 8?
( n 1) n
( n) n 1
1
x
x
3x 3 x
*A good place to start your research is the article by Paul J. Campbell,
“How Old Is the Earth?”, UMAP Modules 1992: Tools for Teaching,
Arlington, MA: Consortium for Mathematics and Its Applications,
1993.
Store f (x) Bx into Y1 and g(x) logBx into Y2 as Experimenting with
different values of B, we find this: For B e1/e 1.444668, the two curves intersect in
two places (where, in terms of B?), for B e1/e they touch only at one place, and for
B e1/e, there is no intersection. (See Classroom Capsules, “An Overlooked Calculus
Question,” The College Mathematics Journal, Vol. 33, No. 5, November 2002.)
ln (x)
ln (B)
Solution for Explore! .
on Page 315
Following Example 4.3.12, store the function f (x) Axe Bx into Y1 of the equation
editor. We attempt to find the maximum of f (x) in terms of A and B. We can set A 1
and vary the value of B (say, 1, 0.5, and 0.01). Then we can fix B to be 0.1 and let
A vary (say, 1, 10, 100).
For instance, when A 1 and B 1, the maximal y value occurs at x 1 (see
the figure on the left). When A 1 and B 0.1, it occurs at x 10 (middle figure).
Solution for Explore!
on Page 333
Complete solutions for all EXPLORE! boxes throughout the text can be accessed at
the book specific website, www.mhhe.com/hoffmann.
One method to display all the desired graphs is to list the desired bases in the function
form. First, write Y1 {1, 2, 3, 4}^X into the equation editor of your graphing
calculator. Observe that for b 1, the functions increase exponentially, with steeper
growth for larger bases. Also all the curves pass through the point (0, 1). Why? Now
try Y1 {2, 4, 6}^X. Note that y 4x lies between y 2x and y 6x. Likewise the
graph of y ex would lie between y 2x and y 3x.
Solution for Explore!
on Page 294
EXPLORE! UPDATE
EXPLORE! UPDATE
4-75 EXPLORE! UPDATE 365
EXPLORE! UPDATE
When A 10 and B 0.1, this maximum remains at x 10 (figure on the right). In
this case, the y coordinate of the maximum increases by the A factor. In general, it
can be shown that the x value of the maximal point is just . The A factor does not
change the location of the x value of the maximum, but it does affect the y value as
a multiplier. To confirm this analytically, set the derivative of y Axe Bx equal to
zero and solve for the location of the maximal point.
Following Example 4.4.2, store f (x) into Y1 and f
(x) into Y2 (but deselected), and
f (x) into Y3 in bold, using the window [ 4.7, 4.7]1 by [ 0.5, 0.5]0.1. Using the
TRACE or the root-finding feature of the calculator, you can determine that the two
x intercepts of f (x) are located at x 1 and x 1. Since the second derivative
f (x) represents the concavity function of f (x), we know that at these values f (x)
changes concavity. At the inflection point ( 1, 0.242), f (x) changes concavity from
positive (concave upward) to negative (concave downward). At x 1, y 0.242, concavity
changes from downward to upward.
Solution for Explore!
on Page 341
1
B
Following Example 4.4.6, store into Y1 and graph using the
window [0, 10]1 by [0, 25]5. We can trace the function for large values of the independent
variable, time. As x approaches 10 (weeks), the function attains a value close
to 20,000 people infected (Y 19.996 thousand). Since 90% of the population is
18,000, by setting Y2 18 and using the intersection feature of the calculator, you can
determine that 90% of the population becomes infected after x 4.28 weeks (about
30 days).
Q(t)
20
Solution for Explore! 1 19e 1.2t
on Page 347
366 CHAPTER 4 Exponential and Logarithmic Functions 4-76
THINK ABOUT IT
FORENSIC ACCOUNTING: BENFORD’S LAW
You might guess that the first digit of each number in a collection of numbers has an
equal chance of being any of the digits 1 through 9, but it was discovered in 1938
by physicist Frank Benford that the chance that the digit is a 1 is more than 30%!
Naturally occurring numbers exhibit a curious pattern in the proportions of the first
digit: smaller digits such as 1, 2, and 3 occur much more often than larger digits, as
seen in the following table:
THINK ABOUT IT
4-77 THINK ABOUT IT 367
First
Digit Proportion
1 30.1%
2 17.6
3 12.5
4 9.7
5 7.9
6 6.7
7 5.8
8 5.1
9 4.6
0.05
1 2 3 4 5 6 7 8 9
0.10
0.15
0.20
0.25
0.30
0.35
Benford distribution
Naturally occurring, in the case, means that the numbers arise without explicit
bound and describe similar quantities, such as the populations of cities or the amounts
paid out on checks. This pattern also holds for exponentially growing numbers and
some types of randomly sampled (but not randomly generated) numbers, and for this
reason it is a powerful tool for determining if these types of data are genuine. The
distribution of digits generally conforms closely to the following rule: the proportion
of numbers such that the first digit is n is given by
This rule, known as Benford’s law, is used to detect fraud in accounting and is
one of several techniques in a field called forensic accounting. Often people who write
fraudulent checks, such as an embezzler at a corporation try to make the first digits
(or even all the digits) occur equally often so as to appear to be random. Benford’s
law predicts that the first digits of such a set of accounting data should not be evenly
proportioned, but rather show a higher occurrence of smaller digits. If an employee
is writing a large number of checks or committing many monetary transfers to a suspicious
target, the check values can be analyzed to see if they follow Benford’s law,
indicating possible fraud if they do not.
log10(n 1) log10 n log10
n 1
n
THINK ABOUT IT
368 CHAPTER 4 Exponential and Logarithmic Functions 4-78
This technique can be applied to various types of accounting data (such as for
taxes and expenses) and has been used to analyze socioeconomic data such as the
gross domestic products of many countries. Benford’s law is also used by the Internal
Revenue Sevice to detect fraud and has been applied to locate errors in data entry
and analysis.
Questions
1. Verify that the formula given for the proportions of digits,
produces the values in the given table. Use calculus to show that the proportion
is a decreasing function of n.
2. The proportions of first digits depend on the base of the number system used.
Computers generally use number systems that are powers of 2. Benford’s law
for base b is
where n ranges from 1 to b. Compute a table like the one given for base b 10
for the bases 4 and 8. Using these computed tables and the given table, do the
proportions seem to be evening out or becoming more uneven as the size of the
base increases?.
Use calculus to justify your assertion by viewing P(n) as a function of b,
for particular values of n. For instance, for n 1:
Use this new function to determine if the proportion of numbers with leading
digit 1 are increasing or decreasing as the size of base b increases. What happens
for the other digits?
f (b) logb 2
ln 2
ln b
P(n) logb
n 1
n
P(n) log10
n 1
n
THINK ABOUT IT
4-79 THINK ABOUT IT 369
3. In the course of a potential fraud investigation, it is found that an employee wrote
checks with the following values to a suspicious source: $234, $444, $513,
$1,120, $2,201, $3,614, $4,311, $5,557, $5,342, $6,710, $8,712, and $8,998.
Compute the proportions corresponding to each of the first digits. Do you think
that fraud may be present? (In actual investigations, statistical tests are used to
determine if the deviation is statistically significant.)
4. Select a collection of numbers arbitrarily from a newspaper or magazine and
record the first digit (the first nonzero digit if it is a decimal less than 1). Do the
numbers appear to follow Benford’s law?
5. The following list of numbers is a sample of heights of prominent mountain
peaks in California, measured in feet. Do they appear to follow Benford’s law?
Source: http://en.wikipedia.org/wiki/Mountain_peaks_of_California.
10,076 1,677 7,196 2,894 9,822
373 1,129 1,558 1,198 343
331 1,119 932 2,563 1,936
1,016 364 1,003 833 765
755 545 1,891 2,027 512
675 2,648 2,601 1,480 719
525 570 884 560 1,362
571 1,992 745 541 385
971 1,220 984 879 1,135
604 2,339 1,588 594 587
References
T. P. Hill, “The First Digit Phenomenon,” American Scientist, Vol. 86, 1998, p. 358.
Steven W. Smith, “The Scientist’s and Engineer’s Guide to Signal Processing,”
chapter 34. Available online at http://www.dspguide.com/ch34/1.htm.
C. Durtschi et al. “The Effective Use of Benfords Law in Detecting Fraud in
Accounting Data.” Availabe online at http://www.auditnet.org/articles/JFA-V-1-
17-34.pdf.
