354
5 Exponential and Logarithmic Functions
Most of the functions we have considered so far have been polynomial
and rational functions, with a few others involving roots or powers of
polynomial or rational functions. The general class of functions defined
by means of the algebraic operations of addition, subtraction, multiplication, division, and the taking of powers and roots on variables and constants are called algebraic functions.
In this chapter we define and investigate the properties of two new and
important types of functions called exponential functions and logarithmic
functions. These functions are not algebraic, but are members of another
class of functions called transcendental functions. The exponential functions and logarithmic functions are used in describing and solving a wide
variety of real-world problems, including growth of populations of people, animals, and bacteria; radioactive decay; growth of money at compound interest; absorption of light as it passes through air, water, or glass;
and magnitudes of sounds and earthquakes. We consider applications in
these areas plus many more in the sections that follow.
SECTION
5-1
Exponential Functions
•
•
•
•
Exponential Functions
Basic Exponential Graphs
Additional Exponential Properties
Applications
In this section we define exponential functions, look at some of their important properties—including graphs—and consider several significant applications.
• Exponential
Functions
Let’s start by noting that the functions f and g given by
f(x) 2x
and
g(x) x2
are not the same function. Whether a variable appears as an exponent with a constant
base or as a base with a constant exponent makes a big difference. The function g is
a quadratic function, which we have already discussed. The function f is a new type
of function called an exponential function.
Many students, if asked to graph an exponential function such as f(x) 2x, would
not hesitate at all. They would likely make up a table by assigning integers to x, plot
the resulting points, and then join these points with a smooth curve, as shown in
Figure 1.
5-1
Exponential Functions
355
y
FIGURE 1 f (x) 2x.
10
5
f (x) 2x
5
5
x
x
f(x)
3
1
8
2
1
4
1
1
2
0
1
1
2
2
4
3
8
The only catch is that we have not defined 2x for all real numbers x. We know
what 25, 23, 22/3, 23/5, 21.4, and 23.15 mean because we have defined 2p for any rational number p, but what does
22
mean? The question is not easy to answer at this time. In fact, a precise definition of
22 must wait for more advanced courses, where we can show that, if b is a positive
real number and x is any real number, then
bx
names a real number, and the graph of f(x) 2x is as indicated in Figure 1. We also
can show that for x irrational, bx can be approximated as closely as we like by using
rational number approximations for x. Since 2 1.414213 . . . , for example, the
sequence
21.4, 21.41, 21.414, . . .
approximates 22, and as we use more decimal places, the approximation improves.
DEFINITION 1
Exponential Function
The equation
f(x) bx
b 0, b 1
defines an exponential function for each different constant b, called the base.
The independent variable x may assume any real value.
Thus, the domain of f is the set of all real numbers, and it can be shown that
the range of f is the set of all positive real numbers. We require the base b to be positive to avoid imaginary numbers such as (2)1/ 2.
356
5 Exponential and Logarithmic Functions
• Basic Exponential
Graphs
EXPLORE-DISCUSS 1
Compare the graphs of f (x) 3x and g(x) 2x by plotting both functions on the
same coordinate system. Find all points of intersection of the graphs. For which
values of x is the graph of f above the graph of g? Below the graph of g? Are the
graphs of f and g close together as x → ? As x → ? Discuss.
It is useful to compare the graphs of y 2x and y (12 ) x 2x by plotting both
on the same coordinate system, as shown in Figure 2(a). The graph of
f(x) bx
b1
[Fig. 2(b)]
looks very much like the graph of the particular case y 2x, and the graph of
f(x) bx
0b1
[Fig. 2(b)]
y
FIGURE 2 Basic exponential
graphs.
y
8
6
4
y 2 2x
1 x
x
4
y bx
b1
y bx
0b1
y 2x
4
x
DOMAIN (, ) RANGE (0, )
(a)
(b)
looks very much like the graph of y Note in both cases that the x axis is a
horizontal asymptote for the graph.
The graphs in Figure 2 suggest the following important general properties of
exponential functions, which we state without proof:
(12) x.
Basic Properties of the Graph of f(x) b x, b 0, b 1
1. All graphs pass through the point (0, 1).
b 0 1 for any permissible base b.
2. All graphs are continuous, with no holes or jumps.
3. The x axis is a horizontal asymptote.
4. If b 1, then bx increases as x increases.
5. If 0 b 1, then bx decreases as x increases.
6. The function f is one-to-one.
5-1
Exponential Functions
357
Property 6 implies that an exponential function has an inverse, called a logarithmic function, which we will discuss in Section 5-3.
A calculator may be used to create an accurate table of values from which the
graph of an exponential function is drawn. Example 1 illustrates the process. (Of
course, we may bypass the creation of a table of values with a graphing utility, which
graphs the function directly.)
EXAMPLE 1
Graphing Multiples of Exponential Functions
Use integer values of x from 3 to 3 to construct a table of values for y 12(4x ), and
then graph this function.
Solution
Use a calculator to create the table of values shown below. Then plot the points, and
join these points with a smooth curve (see Fig. 3).
y
FIGURE 3 y 12 (4x ).
40
30
20
1
10
3
2
• Additional
Exponential
Properties
y
3
0.01
2
0.03
1
0.13
0
0.50
1
2.00
2
8.00
3
32.00
1
5
Matched Problem 1
x
Repeat Example 1 for y
5
12 ( 14 ) x
x
12 (4x ).
Exponential functions, whose domains include irrational numbers, obey the familiar
laws of exponents we discussed earlier for rational exponents. We summarize these
exponent laws here and add two other important and useful properties.
358
5 Exponential and Logarithmic Functions
Exponential Function Properties
For a and b positive, a 1, b 1, and x and y real:
1. Exponent laws:
a xay a xy
a
b
x
ax
bx
(a x) y a xy
ax
a xy
ay
2. a x a y if and only if x y.
(ab) x a xbx
25x
27x
22x
If 64x 62x4, then 4x 2x 4, and x 2.
3. For x 0, then ax bx if and only if a b.
EXAMPLE 2
25x7x
If a4 34, then a 3.
Using Exponential Function Properties
Solve 4x 3 8 for x.
Solution
Express both sides in terms of the same base, and use property 2 to equate exponents.
4x3 8
(22)x3 23
22x6 23
2x 6 3
Express 4 and 8 as powers of 2.
(ax)y axy
Property 2
2x 9
x 92
4(9/2)3 43/2 (4)3 23 ⁄ 8
Check
Matched Problem 2
• Applications
Solve 27x1 9 for x.
We now consider three applications that utilize exponential functions in their analysis: population growth, radioactive decay, and compound interest. Population growth
and compound interest are examples of exponential growth, while radioactive decay
is an example of negative exponential growth.
Our first example involves the growth of populations, such as people, animals,
insects, and bacteria. Populations tend to grow exponentially and at different rates. A
convenient and easily understood measure of growth rate is the doubling time—that
is, the time it takes for a population to double. Over short periods of time the doubling time growth model is often used to model population growth:
P P0 2t/d
5-1
where
Exponential Functions
359
P Population at time t
P0 Population at time t 0
d Doubling time
Note that when t d,
P P02d/d P02
and the population is double the original, as it should be. We use this model to solve
a population growth problem in Example 3.
EXAMPLE 3
Population Growth
Mexico has a population of around 100 million people, and it is estimated that the
population will double in 21 years. If population growth continues at the same rate,
what will be the population:
(A) 15 years from now?
(B) 30 years from now?
Calculate answers to 3 significant digits.
Solutions
We use the doubling time growth model:
P P02 t/d
P (millions)
Substituting P0 100 and d 21, we obtain
500
P 100(2t/21)
400
300
See Figure 4.
(A) Find P when t 15 years:
200
P 100(215/ 21)
100
10
20
30
40
50
164 million people
t
Use a calculator.
Years
FIGURE 4 P 100(2 t/21)
(B) Find P when t 30 years:
P 100(230/ 21)
269 million people
Matched Problem 3
Use a calculator.
The bacterium Escherichia coli (E. coli) is found naturally in the intestines of many
mammals. In a particular laboratory experiment, the doubling time for E. coli is found
to be 25 minutes. If the experiment starts with a population of 1,000 E. coli and there
is no change in the doubling time, how many bacteria will be present:
(A) In 10 minutes?
(B) In 5 hours?
Write answers to 3 significant digits.
360
5 Exponential and Logarithmic Functions
EXPLORE-DISCUSS 2
The doubling time growth model would not be expected to give accurate results
over long periods of time. According to the doubling time growth model of Example 3, what was the population of Mexico 500 years ago at the height of Aztec civilization? What will the population of Mexico be 200 years from now? Explain
why these results are unrealistic. Discuss factors that affect human populations that
are not taken into account by the doubling time growth model.
Our second application involves radioactive decay, which is often referred to as
negative growth. Radioactive materials are used extensively in medical diagnosis and
therapy, as power sources in satellites, and as power sources in many countries. If we
start with an amount A0 of a particular radioactive isotope, the amount declines exponentially in time. The rate of decay varies from isotope to isotope. A convenient and
easily understood measure of the rate of decay is the half-life of the isotope—that is,
the time it takes for half of a particular material to decay. In this section we use the
following half-life decay model:
A A0(12)t /h
A02t /h
where
A Amount at time t
A0 Amount at time t 0
h Half-life
Note that when t h,
A A02h /h A021 A0
2
and the amount of isotope is half the original amount, as it should be.
EXAMPLE 4
Radioactive Decay
The radioactive isotope gallium 67 (67Ga), used in the diagnosis of malignant tumors,
has a biological half-life of 46.5 hours. If we start with 100 milligrams of the isotope, how many milligrams will be left after:
(A) 24 hours?
(B) 1 week?
Compute answers to 3 significant digits.
Solutions
We use the half-life decay model:
A A0(12) t/h A02t/h
Using A0 100 and h 46.5, we obtain
A 100(2t /46.5)
See Figure 5.
5-1
Exponential Functions
361
(A) Find A when t 24 hours:
A (milligrams)
100
A 100(224/46.5)
69.9 milligrams
Use a calculator.
50
(B) Find A when t 168 hours (1 week 168 hours):
100
200
A 100(2168/46.5)
t
8.17 milligrams
Hours
Use a calculator.
FIGURE 5 A 100(2t/46.5).
Matched Problem 4
Radioactive gold 198 (198Au), used in imaging the structure of the liver, has a halflife of 2.67 days. If we start with 50 milligrams of the isotope, how many milligrams
will be left after:
(A)
1
2
day?
(B) 1 week?
Compute answers to 3 significant digits.
Our third application deals with the growth of money at compound interest. This
topic is important to most people and is fundamental to many topics in the mathematics of finance.
The fee paid to use another’s money is called interest. It is usually computed as
a percent, called the interest rate, of the principal over a given period of time. If, at
the end of a payment period, the interest due is reinvested at the same rate, then the
interest earned as well as the principal will earn interest during the next payment
period. Interest paid on interest reinvested is called compound interest.
Suppose you deposit $1,000 in a savings and loan that pays 8% compounded
semiannually. How much will the savings and loan owe you at the end of 2 years?
Compounded semiannually means that interest is paid to your account at the end of
each 6-month period, and the interest will in turn earn interest. The interest rate per
period is the annual rate, 8% 0.08, divided by the number of compounding periods per year, 2. If we let A1, A2, A3, and A4 represent the new amounts due at the end
of the first, second, third, and fourth periods, respectively, then
A1 $1,000 $1,000
2
0.08
$1,000(1 0.04)
A2 A1(1 0.04)
r
n
r
n
2
r
n
3
P 1
[$1,000(1 0.04)](1 0.04)
$1,000(1 0.04)2
A3 A2(1 0.04)
P 1
[$1,000(1 0.04)2](1 0.04)
$1,000(1 0.04)3
P 1
362
5 Exponential and Logarithmic Functions
A4 A3(1 0.04)
[$1,000(1 0.04)3](1 0.04)
$1,000(1 0.04)4
P 1
r
n
4
What do you think the savings and loan will owe you at the end of 6 years? If
you guessed
A $1,000(1 0.04)12
you have observed a pattern that is generalized in the following compound interest
formula:
Compound Interest
If a principal P is invested at an annual rate r compounded n times a year,
then the amount A in the account at the end of t years is given by
AP 1
r
n
nt
The annual rate r is expressed in decimal form.
Since the principal P represents the initial amount in the account and A represents the amount in the account t years later, we also call P the present value of the
account and A the future value of the account.
EXAMPLE 5
Compound Interest
If you deposit $5,000 in an account paying 9% compounded daily, how much will
you have in the account in 5 years? Compute the answer to the nearest cent.
Solution
We use the compound interest formula as follows:
A (dollars)
AP 1
15,000
r
n
nt
5,000 1 10,000
0.09
365
(365)(5)
$7,841.12
5,000
The graph of
5
10
A 5,000 1 t
Years
FIGURE 6
Use a calculator.
is shown in Figure 6.
0.09
365
365 t
5-1
Matched Problem 5
EXAMPLE 6
Exponential Functions
363
If $1,000 is invested in an account paying 10% compounded monthly, how much will
be in the account at the end of 10 years? Compute the answer to the nearest cent.
Visualizing Investments with a Graphing Utility
Use a graphing utility to compare the growth of an investment of $1,000 at 10% compounded monthly with an investment of $2,000 at 5% compounded monthly. When
do the two investments have the same value?
Solution
5000
0
20
We use the compound interest formula to express the future value y1 of the first investment by y1 1,000(1 0.10/12)12x, and the future value y2 of the second investment
by y2 2,000(1 0.05/12)12x, where x is time in years. We graph both functions and
use the intersection routine of the graphing utility to conclude that the investments
have the same value when x 14 years, as shown in Figure 7. After that time the
$1,000 investment has the greater value.
0
FIGURE 7
Matched Problem 6
Use a graphing utility to determine when an investment of $5,000 at 6% compounded
quarterly has the same value as an investment of $4,000 at 10% compounded daily.
Answers to Matched Problems
1. y 12 (4x)
y
40
x
y
3
32.00
2
8.00
1
2.00
0
0.50
1
0.13
2
0.03
3
0.01
30
20
1
10
1
5
2.
3.
4.
5.
6.
x 13
(A) 1,320
(B) 4,100,000 4.10 106
(A) 43.9 mg
(B) 8.12 mg
$2,707.04
5 years, 6 months
5
x
2
3
364
5 Exponential and Logarithmic Functions
EXERCISE
5-1
40. y 7(22x)
A
In Problems 1–10, construct a table of values for integer
values of x over the indicated interval and then graph the
function.
41. g(x) 2x
43. y 1,000(1.08)x
44. y 100(1.03)x
45. y 2x
46. y 3x
2
1. y 3x; [3, 3]
2. y 5x; [2, 2]
C
3. y ( 13 )x 3x; [3, 3]
4. y ( 15 )x 5x; [2, 2]
In Problems 47–50, simplify.
5. g(x) 3x; [3, 3]
6. f(x) 5x; [2, 2]
7. h(x) 5(3 ); [3, 3]
8. f(x) 4(5 ); [2, 2]
x
9. y 3
x3
47. (6x 6x)(6x 6x)
x
5; [6, 0]
10. y 5
x2
12. 52x3564x
4; [4, 0]
7 3yx
14. 4x6y
7
15. (4x)3
17. (102)x (10x)2
18. 2x 2x
20.
32x
73y
3
2
21.
13.
32yx
33x5y
Check Problems 51–54 with a graphing utility.
51. m(x) x(3x)
a2b3c1
a3b2c3
2
53. f(x) 4 a bc
22. a bc 19.
5x
48. (3x 3x)(3x 3x)
Graph each function in Problems 51–54 by constructing a
table of values.
16. (x 3)4
2
2
49. (6x 6x)2 (6x 6x)2 50. (3x 3x)2 (3x 3x)2
In Problems 11–22, simplify.
11. 25x1232x
42. f(x) 2x
2x 2x
2
52. h(x) x(2x)
54. g(x) 3x 3x
2
3
In Problems 55–58:
y
4 3 2
3
6 2 1
B
In Problems 23–34, solve for x.
(A) Approximate the real zeros of each function to two decimal
places.
(B) Investigate the behavior of each function as x → and
x → and find any horizontal asymptotes.
55. f(x) 3x 5
56. f(x) 4 2x
57. f(x) 1 x 10x
58. f(x) 8 x2 2x
23. 32x5 34x2
24. 44x1 42x2
25. 10x 2 102x2
26. 5x
27. (2x 1)3 8
28. (2x 1)5 32
APPLICATIONS
29. 53x 25x3
30. 45x1 162x1
31. 42x2 8x2
32. 1002x4 1,000x4
33. 100x 105x3
34. 3x 9x4
59. Gaming. A person bets on red and black on a roulette
wheel using a Martingale strategy. That is, a $2 bet is
placed on red, and the bet is doubled each time until a win
occurs. The process is then repeated. If black occurs n times
in a row, then L 2n dollars is lost on the nth bet. Graph
this function for 1 n 10. Even though the function is
defined only for positive integers, points on this type of
graph are usually jointed with a smooth curve as a visual
aid.
2
2
2
5
53x5
2
35. Find all real numbers a such that a2 a2. Explain why this
does not violate the second exponential function property
in the box on page 358.
36. Find real numbers a and b such that ab but a4 b4. Explain why this does not violate the third exponential function property in the box on page 358.
Graph each function in Problems 37–46 by constructing a
table of values.
* Check Problems 37–46 with a graphing utility.
37. G(t) 3t/100
38. f(t) 2t/10
39. y 11(3x/2)
*Please note that use of graphing utility is not required to complete
these exercises. Checking them with a g.u. is optional.
60. Bacterial Growth. If bacteria in a certain culture double
every 12 hour, write an equation that gives the number
of bacteria N in the culture after t hours, assuming the culture has 100 bacteria at the start. Graph the equation for
0 t 5.
61. Population Growth. Because of its short life span and frequent breeding, the fruit fly Drosophila is used in some genetic studies. Raymond Pearl of Johns Hopkins University,
5-2
63. Insecticides. The use of the insecticide DDT is no longer
allowed in many countries because of its long-term adverse
effects. If a farmer uses 25 pounds of active DDT, assuming
its half-life is 12 years, how much will still be active after:
(A) 5 years?
(B) 20 years?
Compute answers to 2 significant digits.
64. Radioactive Tracers. The radioactive isotope technetium
99m (99mTc) is used in imaging the brain. The isotope has a
half-life of 6 hours. If 12 milligrams are used, how much
will be present after:
(A) 3 hours?
(B) 24 hours?
Compute answers to 3 significant digits.
SECTION
5-2
365
65. Finance. Suppose $4,000 is invested at 11% compounded
weekly. How much money will be in the account in:
(A) 12 year?
(B) 10 years?
Compute answers to the nearest cent.
for example, studied 300 successive generations of descendants of a single pair of Drosophila flies. In a laboratory situation with ample food supply and space, the doubling time
for a particular population is 2.4 days. If we start with 5
male and 5 female flies, how many flies should we expect to
have in:
(A) 1 week?
(B) 2 weeks?
62. Population Growth. If Kenya has a population of about
30,000,000 people and a doubling time of 19 years and
if the growth continues at the same rate, find the population in:
(A) 10 years
(B) 30 years
Compute answers to 2 significant digits.
The Exponential Function with Base e
66. Finance. Suppose $2,500 is invested at 7% compounded
quarterly. How much money will be in the account in:
(A) 34 year?
(B) 15 years?
Compute answers to the nearest cent.
★
67. Finance. A couple just had a new child. How much should
they invest now at 8.25% compounded daily in order to
have $40,000 for the child’s education 17 years from now?
Compute the answer to the nearest dollar.
★
68. Finance. A person wishes to have $15,000 cash for a new
car 5 years from now. How much should be placed in an account now if the account pays 9.75% compounded weekly?
Compute the answer to the nearest dollar.
★
69. Finance. Will an investment of $10,000 at 8.9% compounded daily ever be worth more at the end of a quarter
than an investment of $10,000 at 9% compounded quarterly? Explain.
★
70. Finance. A sum of $5,000 is invested at 13% compounded
semiannually. Suppose that a second investment of $5,000
is made at interest rate r compounded daily. For which values of r, to the nearest tenth of a percent, is the second investment better than the first? Discuss.
The Exponential Function with Base e
•
•
•
•
Base e Exponential Function
Growth and Decay Applications Revisited
Continuous Compound Interest
A Comparison of Exponential Growth Phenomena
Until now the number has probably been the most important irrational number you
have encountered. In this section we will introduce another irrational number, e, that
is just as important in mathematics and its applications.
• Base e
Exponential Function
The following expression is important to the study of calculus and, as we will see
later in this section, also is closely related to the compound interest formula discussed
in the preceding section:
1
1
m
m