1. Objectives Solve exponential and logarithmic equations and equalities.
Solve problems involving exponential and logarithmic equations.

2. An exponential equation is an equation containing one or more expressions that have a variable as an exponent. To solve exponential equations:
Try writing them so that the bases are all the same.
Take the logarithm of both sides.
When you use a rounded number in a check, the result will not be exact, but it should be reasonable.
Helpful Hint

3. Solve and check.
98 – x = 27x – 3
(32)8 – x = (33)x – 3
Rewrite each side with the same base; 9 and 27 are powers of 3.
316 – 2x = 33x – 9
To raise a power to a power, multiply exponents.
16 – 2x = 3x – 9
Bases are the same, so the exponents must be equal.
x = 5
Solve for x.

4. Check
98 – x = 27x – 3
98 – – 3 
The solution is x = 5.

5. 5 is not a power of 4, so take the log of both sides.
Solve and check.
4x – 1 = 5
log 4x – 1 = log 5
5 is not a power of 4, so take the log of both sides.
(x – 1)log 4 = log 5
Apply the Power Property of Logarithms.
x –1 =
log5
log4
Divide both sides by log 4.
x = ≈ 2.161
log5
log4
Check Use a calculator.
The solution is x ≈

6. Solve and check.
32x = 27
Rewrite each side with the same base; 3 and 27 are powers of 3.
(3)2x = (3)3
32x = 33
To raise a power to a power, multiply exponents.
2x = 3
Bases are the same, so the exponents must be equal.
Check
x = 1.5
Solve for x.
32x = 27
32(1.5) 27
33 27 

7. 21 is not a power of 7, so take the log of both sides.
Check It Out! Example 1b
Solve and check.
7–x = 21
21 is not a power of 7, so take the log of both sides.
log 7–x = log 21
(–x)log 7 = log 21
Apply the Power Property of Logarithms.
–x =
log21
log7
Divide both sides by log 7.
Check
x = – ≈ –1.565
log21
log7

8. 15 is not a power of 2, so take the log of both sides.
Solve and check.
23x = 15
log23x = log15
15 is not a power of 2, so take the log of both sides.
(3x)log 2 = log15
Apply the Power Property of Logarithms.
3x =
log15
log2
Divide both sides by log 2, then divide both sides by 3.
x ≈ 1.302
Check

9. A logarithmic equation is an equation with a logarithmic expression that contains a variable. You can solve logarithmic equations by using the properties of logarithms.
Review the properties of logarithms from Lesson 7-4.
Remember!

10. Use 6 as the base for both sides.
Solve.
log6(2x – 1) = –1
6 log6 (2x –1) = 6–1
Use 6 as the base for both sides.
2x – 1 = 1 6
Use inverse properties to remove 6 to the log base 6. 7 12
x =
Simplify.

11. Use 4 as the base for both sides.
Solve.
log4100 – log4(x + 1) = 1
100
x + 1
log4( ) = 1
Write as a quotient.
4log = 41
100
x + 1
( )
Use 4 as the base for both sides.
= 4
100
x + 1
Use inverse properties on the left side.
x = 24

12. Solve.
log5x 4 = 8
4log5x = 8
Power Property of Logarithms.
log5x = 2
Divide both sides by 4 to isolate log5x.
x = 52
Definition of a logarithm.
x = 25

13. Product Property of Logarithms.
Solve.
log12x + log12(x + 1) = 1
log12 x(x + 1) = 1
Product Property of Logarithms.
log12x(x +1)
= 121
Exponential form.
x(x + 1) = 12
Use the inverse properties.

14. x x2 + x – 12 = 0 Multiply and collect terms. (x – 3)(x + 4) = 0
Factor.
x – 3 = 0 or x + 4 = 0
Set each of the factors equal to zero.
x = 3 or x = –4
Solve.
Check Check both solutions in the original equation.
log12x + log12(x +1) = 1
log12x + log12(x +1) = 1
log123 + log12(3 + 1) 1 x
log12( –4) + log12(–4 +1) 1
log123 + log
log12( –4) is undefined.
log
1 1 
The solution is x = 3.

15. Solve.
3 = log 8 + 3log x
3 = log 8 + 3log x
3 = log 8 + log x3
Power Property of Logarithms.
3 = log (8x3)
Product Property of Logarithms.
103 = 10log (8x3)
Use 10 as the base for both sides.
1000 = 8x3
Use inverse properties on the right side.
125 = x3
5 = x

16. Use 10 as the base for both sides.
Solve.
2log x – log 4 = 0
2log( ) = 0 x 4
Write as a quotient.
2(10log ) = 100 x 4
Use 10 as the base for both sides.
2( ) = 1 x 4
Use inverse properties on the left side.
x = 2

17. Lesson Quiz: Part I
Solve.
x = 5 3
1. 43x–1 = 8x+1
2. 32x–1 = 20
x ≈ 1.86
3. log7(5x + 3) = 3
x = 68
4. log(3x + 1) – log 4 = 2
x = 133
5. log4(x – 1) + log4(3x – 1) = 2
x = 3