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Introduction
So far in this chapter, you have studied the definitions, graphs, and properties of
exponential and logarithmic functions. In this section, you will study procedures
for solving equations involving these exponential and logarithmic functions.
There are two basic strategies for solving exponential or logarithmic
equations. The first is based on the One-to-One Properties and was used to solve
simple exponential and logarithmic equations in Sections 3.1 and 3.2. The
second is based on the Inverse Properties. For and the following
properties are true for all and for which and are defined.
One-to-One Properties
if and only if
if and only if
Inverse Properties
Solving Simple Equations
Original Rewritten
Equation Equation Solution Property
a. One-to-One
b. One-to-One
c. One-to-One
d. Inverse
e. Inverse
f. Inverse
Now try Exercise 13.
The strategies used in Example 1 are summarized as follows.
log x 1 10 log^ x^ 10 ^1 x 10 ^1 101
ln x 3 e ln^ x^ e ^3 x e ^3
e x^ 7 ln e x^ ln 7 x ln 7
^13 3 x^ 3 2 x 2
x
ln x ln 3 0 ln x ln 3 x 3
2 x^ 32 2 x^ 2 5 x 5
log a a x^ x
a log a^ x^ x
log a x log a y x y.
a x^ a y x y.
x y log a x log a y
a > 0 a 1,
246 Chapter 3 Exponential and Logarithmic Functions
What you should learn
- Solve simple exponential and logarithmic equations.
- Solve more complicated exponential equations.
- Solve more complicated logarithmic equations.
- Use exponential and logarith- mic equations to model and solve real-life problems.
Why you should learn it Exponential and logarithmic equations are used to model and solve life science applica- tions. For instance, in Exercise 112, on page 255, a logarithmic function is used to model the number of trees per acre given the average diameter of the trees.
Exponential and Logarithmic Equations
© James Marshall/Corbis
3.
Strategies for Solving Exponential and Logarithmic Equations
1. Rewrite the original equation in a form that allows the use of the
One-to-One Properties of exponential or logarithmic functions.
2. Rewrite an exponential equation in logarithmic form and apply the
Inverse Property of logarithmic functions.
3. Rewrite a logarithmic equation in exponential form and apply the Inverse
Property of exponential functions.
Example 1
Solving Exponential Equations
Solving Exponential Equations
Solve each equation and approximate the result to three decimal places if
necessary.
a. b.
Solution
a. Write original equation.
One-to-One Property Write in general form. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0.
The solutions are and Check these in the original equation.
b. Write original equation.
Divide each side by 3. Take log (base 2) of each side. Inverse Property
Change-of-base formula
The solution is Check this in the original equation.
Now try Exercise 25.
In Example 2(b), the exact solution is and the approximate
solution is An exact answer is preferred when the solution is an
intermediate step in a larger problem. For a final answer, an approximate solution
is easier to comprehend.
Solving an Exponential Equation
Solve and approximate the result to three decimal places.
Solution Write original equation.
Subtract 5 from each side.
Take natural log of each side.
Inverse Property
The solution is ln 55. Check this in the original equation.
Now try Exercise 51.
x 4.
x ln 55 4.
ln e x^ ln 55
e x^ 55
e x^ 5 60
e x^ 5 60
x 3.807.
x log 2 14
x log 2 14 3.807.
x
ln 14
ln 2
x log 2 14
log2 2 x^ log 2 14
2 x^ 14
3 2 x 42
x 1 x 4.
x 4 0 ⇒ x 4
x 1 0 ⇒ x 1
x 1 x 4 0
x^2 3 x 4 0
x^2 3 x 4
e x^2 e ^3 x ^4
e x 3 2 x 42
2
e ^3 x ^4
Section 3.4 Exponential and Logarithmic Equations 247
Remember that the natural loga-
rithmic function has a base of e.
Example 2
Example 3
Solving Logarithmic Equations
To solve a logarithmic equation, you can write it in exponential form.
Logarithmic form
Exponentiate each side.
Exponential form
This procedure is called exponentiating each side of an equation.
Solving Logarithmic Equations
a. Original equation
Exponentiate each side. Inverse Property
b. Original equation
One-to-One Property Add and 1 to each side. Divide each side by 4.
c. Original equation
Quotient Property of Logarithms
One-to-One Property
Cross multiply. Isolate Divide each side by
Now try Exercise 77.
Solving a Logarithmic Equation
Solve and approximate the result to three decimal places.
Solution Write original equation.
Subtract 5 from each side.
Divide each side by 2.
Exponentiate each side.
Inverse Property
Use a calculator.
Now try Exercise 85.
x 0.
x e ^1 ^2
e ln^ x^ e ^1 ^2
ln x
2 ln x 1
5 2 ln x 4
5 2 ln x 4
x 2 7.
7 x 14 x.
3 x 14 10 x
3 x 14
2 x
log6
3 x 14
5 ^
log 6 2 x
log 6 3 x 14 log 6 5 log6 2 x
x 2
4 x 8 x
5 x 1 x 7
log 3 5 x 1 log3 x 7
x e^2
e ln^ x^ e^2
ln x 2
x e^3
e ln^ x^ e^3
ln x 3
Section 3.4 Exponential and Logarithmic Equations 249
Example 6
Example 7
Remember to check your solu-
tions in the original equation
when solving equations to verify
that the answer is correct and to
make sure that the answer lies
in the domain of the original
equation.
Activities
- Solve for
Answer:
- Solve for
Answer: is not in the domain.
x 1 x 6
logx 4 logx 1 1.
x:
x
ln 3 ln 7 ^ 0.
x: 7 x^ 3.
Solving a Logarithmic Equation
Solve 2
Solution Write original equation.
Divide each side by 2.
Exponentiate each side (base 5).
Inverse Property
Divide each side by 3.
The solution is Check this in the original equation.
Now try Exercise 87.
Because the domain of a logarithmic function generally does not include all
real numbers, you should be sure to check for extraneous solutions of logarithmic
equations.
x 253.
x
3 x 25
5 log^5 3 x^ 52
log5 3 x 2
2 log 5 3 x 4
log5 3 x 4.
250 Chapter 3 Exponential and Logarithmic Functions
Example 8
Notice in Example 9 that the
logarithmic part of the equation
is condensed into a single loga-
rithm before exponentiating
each side of the equation.
In Example 9, the domain of is and the domain of is
so the domain of the original equation is Because the domain is all
real numbers greater than 1, the solution is extraneous. The graph in
Figure 3.26 verifies this concept.
x 4
x > 1, x > 1.
log 5 x x > 0 log x 1
Checking for Extraneous Solutions
Solve log 5 x log x 1 2.
Example 9
Algebraic Solution
Write original equation.
Product Property of Logarithms
Exponentiate each side (base 10).
Inverse Property
Write in general form.
Factor.
Set 1st factor equal to 0.
Solution
Set 2nd factor equal to 0.
Solution
The solutions appear to be and However, when
you check these in the original equation, you can see that
is the only solution.
Now try Exercise 99.
x 5
x 5 x 4.
x 4
x 4 0
x 5
x 5 0
x 5 x 4 0
x^2 x 20 0
5 x^2 5 x 100
10 log^5 x
(^2) 5 x
log 5 x x 1 2
log 5 x log x 1 2
Graphical Solution
Use a graphing utility to graph
and in the same
viewing window. From the graph shown in Figure
3.26, it appears that the graphs intersect at one point.
Use the intersect feature or the zoom and trace
features to determine that the graphs intersect at
approximately So, the solution is
Verify that 5 is an exact solution algebraically.
FIGURE 3.
0
− 1
9
y 1 = log 5 x + log( x − 1)
y 2 = 2
5
5, 2. x 5.
y 1 log 5 x log x 1 y 2 2
WRITING ABOUT^ MATHEMATICS
Endangered Animals
The number of endangered animal species in the United States from 1990 to
2002 can be modeled by
where represents the year, with corresponding to 1990 (see Figure 3.28).
During which year did the number of endangered animal species reach 357?
(Source: U.S. Fish and Wildlife Service)
Solution Write original equation.
Substitute 357 for
Add 119 to each side.
Divide each side by 164.
Exponentiate each side.
Inverse Property
Use a calculator.
The solution is Because represents 1990, it follows that the
number of endangered animals reached 357 in 1998.
Now try Exercise 113.
t 18. t 10
t 18
t e^476 ^164
e ln^ t^ e^476 ^164
ln t
164 ln t 476
119 164 ln t 357 y.
119 164 ln t y
t t 10
y 119 164 ln t , 10 ≤ t ≤ 22
y
252 Chapter 3 Exponential and Logarithmic Functions
Example 11
10 12 14 16 18 20 22
200
250
300
350
400
450
t
y
Endangered Animal Species
Number of species
Year (10 ↔ 1990)
FIGURE 3.
Comparing Mathematical Models The table shows the U.S.
Postal Service rates y for sending an express mail package for selected years from 1985 through 2002, where
represents 1985. (Source: U.S. Postal Service)
a. Create a scatter plot of the data. Find a linear model for the data, and add its graph to your scatter plot. According to this model, when will the rate for sending an express mail package reach $19.00? b. Create a new table showing values for ln x and ln y and create a scatter plot of these transformed data. Use the method illustrated in Example 7 in Section 3.3 to find a model for the transformed data, and add its graph to your scatter plot. According to this model, when will the rate for sending an express mail package reach $19.00? c. Solve the model in part (b) for y, and add its graph to your scatter plot in part (a). Which model better fits the original data? Which model will better predict future rates? Explain.
x 5
Year, x Rate, y
Section 3.4 Exponential and Logarithmic Equations 253
3.4 Exercises
In Exercises 1–8, determine whether each -value is a solution (or an approximate solution) of the equation.
1. 2. (a) (a) (b) (b) 3. (a) (b) (c) 4. (a)
(b)
(c) 5. (a) (b) (c) 6. (a) (b) (c) 7. (a) (b) (c) 8. (a) (b) (c)
In Exercises 9–20, solve for
**9. 10.
- 20.**
In Exercises 21–24, approximate the point of intersection of the graphs of and Then solve the equation algebraically to verify your approximation.
21. 22.
x
−
8 12
4
8
12
f g
y
f x
g
y
4 8 12
4
g x 2 g x 0
f x log 3 x f x ln x 4
x −8 − −
4 8
4
8
12
f
g
y
x −8 − −
4 8
4
12
f
g
y
g x 8 g x 9
f x 2 x f x 27 x
f x g x
f g.
log 4 x 3 log 5 x 3
ln x 1 ln x 7
e x^ 2 e x^ 4
ln x ln 2 0 ln x ln 5 0
^14
x ^12 ^ ^64
x 32
4 x^ 16 3 x^ 243
x.
x 1 ln 3.
x 45.
x 1 e 3.
ln x 1 3.
x 163.
x 12 3 e 5.8
x 12 3 ln 5.8
ln 2 x 3 5.
x 10 2 3
x 17
x 1021
log 2 x 3 10
x (^643)
x 4
x 21.
log 4 3 x 3
x 0.
x
ln 6 5 ln 2
x 15 2 ln 6
2 e^5 x ^2 12
x 1.
x 2 ln 25
x 2 e^25
3 e x ^2 75
x 2 x 2
x 5 x 1
4 2 x ^7 64 2 3 x ^1 32
x
VOCABULARY CHECK: Fill in the blanks.
1. To ________ an equation in means to find all values of for which the equation is true. 2. To solve exponential and logarithmic equations, you can use the following One-to-One and Inverse Properties. (a) if and only if ________. (b) if and only if ________. (c) ________ (d) ________ 3. An ________ solution does not satisfy the original equation.
PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.
log a a x^
a log a^ x^
log a x log a y
ax^ ay
x x
Section 3.4 Exponential and Logarithmic Equations 255
112. Trees per Acre The number of trees of a given species per acre is approximated by the model where is the average diameter of the trees (in inches) 3 feet above the ground. Use the model to approximate the average diameter of the trees in a test plot when 113. Medicine The number of hospitals in the United States from 1995 to 2002 can be modeled by
where represents the year, with corresponding to
- During which year did the number of hospitals reach 5800? (Source: Health Forum) 114. Sports The number of daily fee golf facilities in the United States from 1995 to 2003 can be modeled by where represents the year, with corresponding to 1995. During which year did the number of daily fee golf facilities reach 9000? (Source: National Golf Foundation) 115. Average Heights The percent of American males between the ages of 18 and 24 who are no more than inches tall is modeled by
and the percent of American females between the ages of 18 and 24 who are no more than inches tall is modeled by
(Source: U.S. National Center for Health Statistics) (a) Use the graph to determine any horizontal asymptotes of the graphs of the functions. Interpret the meaning in the context of the problem.
(b) What is the average height of each sex?
116. Learning Curve In a group project in learning theory, a mathematical model for the proportion of correct responses after trials was found to be
(a) Use a graphing utility to graph the function. (b) Use the graph to determine any horizontal asymptotes of the graph of the function. Interpret the meaning of the upper asymptote in the context of this problem. (c) After how many trials will 60% of the responses be correct?
P
1 e 0.2 n^
n
P
Height (in inches)
Percent ofpopulation
x
f ( x )
m ( x ) 20
40
60
80
100
55 60 65 70 75
f x
1 e 0.66607 x 64.51
x
f
m x
1 e 0.6114 x 69.71
x
m
t 5
y 4381 1883.6 ln t , 5 ≤ t ≤ 13 t
y
t t 5
y 7312 630.0 ln t , 5 ≤ t ≤ 12
y
N 21.
N 68 10 0.04 x , 5 ≤ x ≤ 40 x
N
x 0.2 0.4 0.6 0.8 1.
y
117. Automobiles Automobiles are designed with crum- ple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer g’s the crash victims experience. (One g is equal to the acceleration due to gravity. For very short periods of time, humans have withstood as much as 40 g’s.) In crash tests with vehicles moving at 90 kilometers per hour, analysts measured the numbers of g’s experienced during deceleration by crash dummies that were permitted to move meters during impact. The data are shown in the table.
A model for the data is given by
where is the number of g’s. (a) Complete the table using the model.
(b) Use a graphing utility to graph the data points and the model in the same viewing window. How do they compare? (c) Use the model to estimate the distance traveled during impact if the passenger deceleration must not exceed 30 g’s. (d) Do you think it is practical to lower the number of g’s experienced during impact to fewer than 23? Explain your reasoning.
y
y 3.00 11.88 ln x
x
x
Model It
x g’s
118. Data Analysis An object at a temperature of 160 C was removed from a furnace and placed in a room at 20 C. The temperature of the object was measured each hour and recorded in the table. A model for the data is given by The graph of this model is shown in the figure.
(a) Use the graph to identify the horizontal asymptote of the model and interpret the asymptote in the context of the problem. (b) Use the model to approximate the time when the temperature of the object was 100 C.
Synthesis
True or False? In Exercises 119–122, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer.
119. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. 120. The logarithm of the sum of two numbers is equal to the product of the logarithms of the numbers. 121. The logarithm of the difference of two numbers is equal to the difference of the logarithms of the numbers. 122. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers. 123. Think About It Is it possible for a logarithmic equation to have more than one extraneous solution? Explain. 124. Finance You are investing dollars at an annual interest rate of compounded continuously, for years. Which of the following would result in the highest value of the investment? Explain your reasoning. (a) Double the amount you invest. (b) Double your interest rate. (c) Double the number of years. 125. Think About It Are the times required for the invest- ments in Exercises 107 and 108 to quadruple twice as long as the times for them to double? Give a reason for your answer and verify your answer algebraically. 126. Writing Write two or three sentences stating the general guidelines that you follow when solving (a) exponential equations and (b) logarithmic equations.
Skills Review
In Exercises 127–130, simplify the expression.
**127.
129.**
130.
In Exercises 131–134, sketch a graph of the function.
131. 132.
133.
In Exercises 135–138, evaluate the logarithm using the change-of-base formula. Approximate your result to three decimal places.
**135.
137.**
138. log 8 22
log 3 4 5
log 3 4
log 6 9
g x
x 3, x^2 1,
x ≤ 1 x > 1
g x
2 x , x^2 4,
x < 0 x ≥ 0
f x x 2 8
f x x 9
(^3 25) 3 15
48 x^2 y^5
r , t
P
T
h
Temperature (in degrees Celsius) 20
40
60
80
100
120
140
160
1 2 3 4 5 6 7 8 Hour
T 20 ^1 ^7 ^2 h .
h
T
256 Chapter 3 Exponential and Logarithmic Functions
Hour, h Temperature, T
