Section 6.5 Solving Exponential Equations 325
6.5
Essential QuestionEssential Question How can you solve an exponential equation
graphically?
Solving an Exponential Equation Graphically
Work with a partner. Use a graphing calculator to solve the exponential equation
2.5
x − 3
= 6.25 graphically. Describe your process and explain how you determined
the solution.
The Number of Solutions of an
Exponential Equation
Work with a partner.
a. Use a graphing calculator to graph the equation y = 2
x
.
6
−2
−6
6
b. In the same viewing window, graph a linear equation (if possible) that does not
intersect the graph of y = 2
x
.
c. In the same viewing window, graph a linear equation (if possible) that intersects the
graph of y = 2
x
in more than one point.
d. Is it possible for an exponential equation to have no solution? more than one
solution? Explain your reasoning.
Solving Exponential Equations Graphically
Work with a partner. Use a graphing calculator to solve each equation.
a. 2
x
=
1
—
2
b. 2
x + 1
= 0 c. 2
x
=
√
—
2
d. 3
x
= 9 e. 3
x − 1
= 0 f. 4
2x
= 2
g. 2
x/2
=
1
—
4
h. 3
x + 2
=
1
—
9
i. 2
x − 2
=
3
—
2
x − 2
Communicate Your AnswerCommunicate Your Answer
4. How can you solve an exponential equation graphically?
5. A population of 30 mice is expected to double each year. The number p of mice in
the population each year is given by p
=
30(2
n
). In how many years will there be
960 mice in the population?
USING
APPROPRIATE
TOOLS
To be pro cient in
math, you need to use
technological tools to
explore and deepen
your understanding
of concepts.
Solving Exponential Equations
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326 Chapter 6 Exponential Functions and Sequences
6.5
Lesson
Property of Equality for Exponential Equations
Words Two powers with the same positive base b, where b ≠ 1, are equal if and
only if their exponents are equal.
Numbers If 2
x
= 2
5
, then x
=
5. If x
=
5, then 2
x
= 2
5
.
Algebra If b > 0 and b ≠ 1, then b
x
= b
y
if and only if x
=
y.
What You Will LearnWhat You Will Learn
Solve exponential equations with the same base.
Solve exponential equations with unlike bases.
Solve exponential equations by graphing.
Solving Exponential Equations with the Same Base
Exponential equations are equations in which variable expressions occur
as exponents.
exponential equation, p. 326
Core Vocabulary
Core Vocabu
l
l
a
r
r
y
Solving Exponential Equations with the Same Base
Solve each equation.
a. 3
x + 1
= 3
5
b. 6 = 6
2x − 3
c. 10
3x
= 10
2x + 3
SOLUTION
a. 3
x + 1
= 3
5
Write the equation.
x + 1 = 5 Equate the exponents.
− 1 − 1 Subtract 1 from each side.
x = 4 Simplify.
b. 6 = 6
2x − 3
Write the equation.
1 = 2x − 3 Equate the exponents.
+ 3 + 3 Add 3 to each side.
4 = 2x Simplify.
4
—
2
=
2x
—
2
Divide each side by 2.
2 = x Simplify.
c. 10
3x
= 10
2x + 3
Write the equation.
3x = 2x + 3 Equate the exponents.
− 2x − 2x Subtract 2
x
from each side.
x = 3 Simplify.
Monitoring ProgressMonitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the equation. Check your solution.
1. 2
2x
= 2
6
2. 5
2x
= 5
x + 1
3. 7
3x + 5
= 7
x + 1
Check
6 = 6
2x − 3
6 =
?
6
2(2) − 3
6 = 6 ✓
Core Core ConceptConcept
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Section 6.5 Solving Exponential Equations 327
Solving Exponential Equations with Unlike Bases
To solve some exponential equations, you must  rst rewrite each side of the equation
using the same base.
Solving Exponential Equations with Unlike Bases
Solve (a) 5
x
= 125, (b) 4
x
= 2
x − 3
, and (c) 9
x + 2
= 27
x
.
SOLUTION
a. 5
x
= 125 Write the equation.
5
x
= 5
3
Rewrite 125 as 5
3
.
x = 3 Equate the exponents.
b. 4
x
= 2
x − 3
Write the equation.
(2
2
)
x
= 2
x − 3
Rewrite 4 as 2
2
.
2
2x
= 2
x − 3
Power of a Power Property
2x = x − 3 Equate the exponents.
x = −3 Solve for
x
.
c. 9
x + 2
= 27
x
Write the equation.
(3
2
)
x + 2
= (3
3
)
x
Rewrite 9 as 3
2
and 27 as 3
3
.
3
2x + 4
= 3
3x
Power of a Power Property
2x + 4 = 3x Equate the exponents.
4 = x Solve for
x
.
Solving Exponential Equations When 0 < b < 1
Solve (a)
(
1
—
2
)
x
= 4 and (b) 4
x + 1
=
1
—
64
.
SOLUTION
a.
(
1
—
2
)
x
= 4 Write the equation.
(2
−1
)
x
= 2
2
Rewrite
1
—
2
as 2
−
1
and 4 as 2
2
.
2
−x
= 2
2
Power of a Power Property
−x = 2 Equate the exponents.
x = −2 Solve for
x
.
b. 4
x + 1
=
1
—
64
Write the equation.
4
x + 1
=
1
—
4
3
Rewrite 64 as 4
3
.
4
x + 1
= 4
−3
De nition of negative exponent
x + 1 = −3 Equate the exponents.
x = −4 Solve for
x
.
Monitoring ProgressMonitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the equation. Check your solution.
4. 4
x
= 256 5. 9
2x
= 3
x − 6
6. 4
3x
= 8
x + 1
7.
(
1
—
3
)
x − 1
= 27
Check
4
x
= 2
x − 3
4
−3
=
?
2
−3 − 3
1
—
64
=
1
—
64
✓
Check
9
x + 2
= 27
x
9
4 + 2
=
?
27
4
531,441 = 531,441 ✓
Check
4
x + 1
=
1
—
64
4
−4 + 1
=
?
1
—
64
1
—
64
=
1
—
64
✓
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328 Chapter 6 Exponential Functions and Sequences
Solving Exponential Equations by Graphing
Sometimes, it is impossible to rewrite each side of an exponential equation using the
same base. You can solve these types of equations by graphing each side and  nding
the point(s) of intersection. Exponential equations can have no solution, one solution,
or more than one solution depending on the number of points of intersection.
Solving Exponential Equations by Graphing
Use a graphing calculator to solve (a)
(
1
—
2
)
x − 1
= 7 and (b) 3
x + 2
= x + 1.
SOLUTION
a. Step 1 Write a system of equations using each side of the equation.
y =
(
1
—
2
)
x − 1
Equation 1
y = 7 Equation 2
Step 2 Enter the equations into a calculator.
Then graph the equations in a viewing
window that shows where the graphs
could intersect.
Step 3 Use the intersect feature to  nd the
point of intersection. Thegraphs
intersect at about (−1.81, 7).
So, the solution is x ≈ −1.81.
b. Step 1 Write a system of equations using each side of the equation.
y = 3
x + 2
Equation 1
y = x + 1 Equation 2
Step 2 Enter the equations into a calculator.
Then graph the equations in a
viewing window that shows where
the graphs could intersect.
The graphs do not intersect. So, the equation has no solution.
Monitoring ProgressMonitoring Progress
Help in English and Spanish at BigIdeasMath.com
Use a graphing calculator to solve the equation.
8. 2
x
= 1.8 9. 4
x − 3
= x + 2 10.
(
1
—
4
)
x
= −2x − 3
Check
(
1
—
2
)
x − 1
= 7
(
1
—
2
)
−1.81 − 1
=
?
7
7.01 ≈ 7 ✓
−10
−10
10
10
−10
−10
10
10
Intersection
X=-1.807355
Y=7
−10
−10
10
10
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Section 6.5 Solving Exponential Equations 329
Dynamic Solutions available at BigIdeasMath.com
Exercises
6.5
In Exercises 3–12, solve the equation. Check your
solution. (See Examples 1 and 2.)
3. 4
5x
= 4
10
4. 7
x − 4
= 7
8
5. 3
9x
= 3
7x + 8
6. 2
4x
= 2
x + 9
7. 2
x
= 64 8. 3
x
= 243
9. 7
x − 5
= 49
x
10. 216
x
= 6
x + 10
11. 64
2x + 4
= 16
5x
12. 27
x
= 9
x − 2
In Exercises 13–18, solve the equation. Check your
solution. (See Example 3.)
13.
(
1
—
5
)
x
= 125 14.
(
1
—
4
)
x
= 256
15.
1
—
128
= 2
5x + 3
16. 3
4x − 9
=
1
—
243
17. 36
−3x + 3
=
(
1
—
216
)
x + 1
18.
(
1
—
27
)
4 − x
= 9
2x − 1
ERROR ANALYSIS In Exercises 19 and 20, describe and
correct the error in solving the exponential equation.
19.
5
3x 2
25
x 8
3x 2 x 8
x 5
✗
20.
(
1
—
8
)
5x
32
x 8
(2
3
)
5x
(2
5
)
x 8
2
15x
2
5x 40
15x 5x 40
x 4
✗
In Exercises 21–24, match the equation with the graph
that can be used to solve it. Then solve the equation.
21. 2
x
= 6 22. 4
2x − 5
= 6
23. 5
x + 2
= 6 24. 3
−x − 1
= 6
A. B.
−2
−55
8
Intersection
X=-2.63093 Y=6
−2
−55
8
Intersection
X=-.8867172 Y=6
C. D.
−2
−55
8
Intersection
X=2.5849625 Y=6
−2
−55
8
Intersection
X=3.1462406 Y=6
In Exercises 25–36, use a graphing calculator to solve
the equation. (See Example 4.)
25. 6
x + 2
= 12 26. 5
x − 4
= 8
27.
(
1
—
2
)
7x + 1
= −9 28.
(
1
—
3
)
x + 3
= 10
29. 2
x + 6
= 2x + 15
30. 3x
− 2
= 5
x − 1
31.
1
—
2
x − 1 =
(
1
—
3
)
2x − 1
32. 2
−x + 1
= −
3
—
4
x
+ 3
33. 5
x
= −4
−x + 4
34. 7
x − 2
= 2
−x
35. 2
−x − 3
= 3
x + 1
36. 5
−2x + 3
= −6
x + 5
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. WRITING Describe how to solve an exponential equation with unlike bases.
2. WHICH ONE DOESN’T BELONG? Which equation does not belong with the other three?
Explain your reasoning.
2
x
= 4
x + 6
3
4
= x + 4
2
5
3x + 8
= 5
2x
2
x − 7
= 2
7
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
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330 Chapter 6 Exponential Functions and Sequences
In Exercises 37–40, solve the equation by using the
Property of Equality for Exponential Equations.
37. 30
⋅
5
x + 3
= 150 38. 12
⋅
2
x − 7
= 24
39. 4(3
−2x − 4
) = 36 40. 2(4
2x + 1
) = 128
41. MODELING WITH MATHEMATICS You scan a photo
into a computer at four times its original size. You
continue to increase its size repeatedly by 100%
using the computer. The new size of the photo y in
comparison to its original size after x enlargements on
the computer is represented by y = 2
x + 2
. How many
times must the photo be enlarged on the computer so
the new photo is 32 times the original size?
42. MODELING WITH MATHEMATICS A bacterial culture
quadruples in size every hour. You begin observing
the number of bacteria 3 hours after the culture is
prepared. The amount y of bacteria x hours after the
culture is prepared is represented by y = 192(4
x − 3
).
When will there be 200,000 bacteria?
In Exercises 43–46, solve the equation.
43. 3
3x + 6
= 27
x + 2
44. 3
4x + 3
= 81
x
45. 4
x + 3
= 2
2(x + 1)
46. 5
8(x − 1)
= 625
2x − 2
47. NUMBER SENSE Explain how you can use mental
math to solve the equation 8
x − 4
=
1.
48. PROBLEM SOLVING There are a total of 128 teams at
the start of a citywide 3-on-3 basketball tournament.
Half the teams are eliminated after each round. Write
and solve an exponential equation to determine after
which round there are 16 teams left.
49. PROBLEM SOLVING You deposit $500 in a savings
account that earns 6% annual interest compounded
yearly. Write and solve an exponential equation
to determine when the balance of the account will
be $800.
50. HOW DO YOU SEE IT? The graph shows the annual
attendance at two different events. Each event began
in 2004.
0
4000
8000
12,000
Number of people
Year (0 ↔ 2004)
24680 x
y
Event Attendance
y = 4000(1.25)
x
y = 12,000(0.87)
x
Event 1
Event 2
a. Estimate when the events will have about the
sameattendance.
b. Explain how you can verify your answer in
part(a).
51. REASONING Explain why the Property of Equality
for Exponential Equations does not work when b = 1.
Give an example to justify your answer.
52. THOUGHT PROVOKING Is it possible for an
exponential equation to have two different solutions?
If not, explain your reasoning. If so, give an example.
USING STRUCTURE In Exercises 53– 58, solve the equation.
53. 8
x − 2
=
√
—
8 54.
√
—
5 = 5
x + 4
55.
(
5
√
—
7
)
x
= 7
2x + 3
56. 12
2x − 1
=
(
4
√
—
12
)
x
57.
(
3
√
—
6
)
2x
=
(
√
—
6
)
x + 6
58.
(
5
—
3
)
5x − 10
=
(
8
√
—
3
)
4x
59. MAKING AN ARGUMENT Consider the equation
(
1
—
a
)
x
= b, where a > 1 and b > 1. Your friend says
the value of x will always be negative. Is your friend
correct? Explain.
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Determine whether the sequence is arithmetic. If so,  nd the common difference. (Section 4.6)
60. −20, −26, −32, −38, . . . 61. 9, 18, 36, 72, . . .
62. −5, −8, −12, −17, . . . 63. 10, 20, 30, 40, . . .
Reviewing what you learned in previous grades and lessons
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