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7-5
Exponential and Logarithmic
Equations and Inequalities
Objectives
Solve exponential and logarithmic
equations and equalities.
Solve problems involving exponential
and logarithmic equations.
Holt Algebra 2
7-5
Exponential and Logarithmic
Equations and Inequalities
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.
Helpful Hint
When you use a rounded number in a check, the
result will not be exact, but it should be
reasonable.
Holt Algebra 2
7-5
Exponential and Logarithmic
Equations and Inequalities
Solve and check.
98 – x = 27x – 3
(32)8 – x = (33)x – 3
316 – 2x = 33x – 9
16 – 2x = 3x – 9
x=5
Holt Algebra 2
Rewrite each side with the same
base; 9 and 27 are powers of 3.
To raise a power to a power,
multiply exponents.
Bases are the same, so the
exponents must be equal.
Solve for x.
7-5
Exponential and Logarithmic
Equations and Inequalities
Check
98 – x = 27x – 3
98 – 5 275 – 3
93 272
729 729 
The solution is x = 5.
Holt Algebra 2
7-5
Exponential and Logarithmic
Equations and Inequalities
Solve and check.
4x – 1 = 5
log 4x – 1 = log 5
(x – 1)log 4 = log 5
log5
x –1 = log4
5 is not a power of 4, so take the
log of both sides.
Apply the Power Property of
Logarithms.
Divide both sides by log 4.
log5
x = 1 + log4 ≈ 2.161
The solution is x ≈ 2.161.
Holt Algebra 2
Check Use a calculator.
7-5
Exponential and Logarithmic
Equations and Inequalities
Solve and check.
32x = 27
2x
(3)
3
= (3)
Rewrite each side with the same
base; 3 and 27 are powers of 3.
32x = 33
To raise a power to a power,
multiply exponents.
2x = 3
Bases are the same, so the
exponents must be equal.
x = 1.5
Solve for x.
Check
32x =
27
32(1.5) 27
33 27
27 27 
Holt Algebra 2
7-5
Exponential and Logarithmic
Equations and Inequalities
Check It Out! Example 1b
Solve and check.
7–x = 21
log 7–x = log 21
(–x)log 7 = log 21
log21
–x = log7
log21
21 is not a power of 7, so take the
log of both sides.
Apply the Power Property of
Logarithms.
Divide both sides by log 7.
x = – log7 ≈ –1.565
Holt Algebra 2
Check
Exponential and Logarithmic
7-5 Equations and Inequalities
Solve and check.
23x = 15
log23x = log15
(3x)log 2 = log15
log15
3x = log2
x ≈ 1.302
Holt Algebra 2
15 is not a power of 2, so take the
log of both sides.
Apply the Power Property of
Logarithms.
Divide both sides by log 2,
then divide both sides by 3.
Check
7-5
Exponential and Logarithmic
Equations and Inequalities
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.
Remember!
Review the properties of logarithms from Lesson
7-4.
Holt Algebra 2
7-5
Exponential and Logarithmic
Equations and Inequalities
Solve.
log6(2x – 1) = –1
6
log (2x –1)
6
= 6–1
2x – 1 = 1
6
7
x = 12
Holt Algebra 2
Use 6 as the base for both sides.
Use inverse properties to remove
6 to the log base 6.
Simplify.
7-5
Exponential and Logarithmic
Equations and Inequalities
Solve.
log4100 – log4(x + 1) = 1
100
log4(x + 1 ) = 1
log4( x + 1 )
100
4
= 41
100
=4
x+1
x = 24
Holt Algebra 2
Write as a quotient.
Use 4 as the base for both sides.
Use inverse properties on the
left side.
7-5
Exponential and Logarithmic
Equations and Inequalities
Solve.
log5x 4 = 8
4log5x = 8
log5x = 2
x = 52
x = 25
Holt Algebra 2
Power Property of Logarithms.
Divide both sides by 4 to isolate log5x.
Definition of a logarithm.
7-5
Exponential and Logarithmic
Equations and Inequalities
Solve.
log12x + log12(x + 1) = 1
log12 x(x + 1) = 1
12 log
x(x +1)
12
= 121
x(x + 1) = 12
Holt Algebra 2
Product Property of Logarithms.
Exponential form.
Use the inverse properties.
7-5
Exponential and Logarithmic
Equations and Inequalities
x2 + x – 12 = 0
(x – 3)(x + 4) = 0
Multiply and collect terms.
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)
log123 + log124
log1212
1
1
1
1
1
log12( –4) + log12(–4 +1) 1 x
log12( –4) is undefined.
The solution is x = 3.
Holt Algebra 2
7-5
Exponential and Logarithmic
Equations and Inequalities
Solve.
3 = log 8 + 3log x
3 = log 8 + 3log x
3 = log 8 + log x3
3 = log (8x3)
103 = 10log (8x3)
1000 = 8x3
125 = x3
5=x
Holt Algebra 2
Power Property of Logarithms.
Product Property of Logarithms.
Use 10 as the base for both sides.
Use inverse properties on the
right side.
7-5
Exponential and Logarithmic
Equations and Inequalities
Solve.
2log x – log 4 = 0
x
2log( 4
log
2(10
)=0
Write as a quotient.
x
4
Use 10 as the base for both sides.
) = 100
2( x ) = 1
4
x=2
Holt Algebra 2
Use inverse properties on the
left side.
7-5
Exponential and Logarithmic
Equations and Inequalities
Lesson Quiz: Part I
Solve.
1. 43x–1 = 8x+1
x= 5
3
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
Holt Algebra 2