Holt Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve exponential and logarithmic equations and equalities.
Solve problems involving exponential and logarithmic equations.
Objectives
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.
When you use a rounded number in a check, the result will not be exact, but it should be reasonable.
Helpful Hint
Holt Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
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.
Holt Algebra 2
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 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.
Divide both sides by log 4.
Check Use a calculator.
The solution is x ≈ 2.161.
x = 1 + ≈ 2.161log5log4
x –1 = log5log4
Holt Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve and check.
32x = 27
(3)2x = (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) 2733 27
27 27
Holt Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve and check.
7–x = 21
log 7–x = log 21 21 is not a power of 7, so take the log of both sides.
(–x)log 7 = log 21 Apply the Power Property of Logarithms.
Check It Out! Example 1b
Divide both sides by log 7.
x = – ≈ –1.565log21log7
–x = log21log7
Check
Holt Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
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.
Divide both sides by log 2, then divide both sides by 3.
x ≈ 1.302
3x = log15 log2
Check
Holt Algebra 2
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.
Review the properties of logarithms from Lesson 7-4.
Remember!
Holt Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve.
Use 6 as the base for both sides.
log6(2x – 1) = –1
6 log6
(2x –1) = 6–1
2x – 1 = 1 6
7 12
x =
Use inverse properties to remove 6 to the log base 6.
Simplify.
Holt Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve.
Write as a quotient.
log4100 – log4(x + 1) = 1
x = 24
Use 4 as the base for both sides.
Use inverse properties on the left side.
100 x + 1log
4( ) = 1
4log4 = 41
100x + 1( )
= 4 100 x + 1
Holt Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve.
Power Property of Logarithms.
log5x 4 = 8
x = 25
Definition of a logarithm.
4log5x = 8
log5x = 2
x = 52
Divide both sides by 4 to isolate log5x.
Holt Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve.
Product Property of Logarithms.
log12
x + log12
(x + 1) = 1
Exponential form.
Use the inverse properties.
log12
x(x + 1) = 1
log12
x(x +1) 12 = 121
x(x + 1) = 12
Holt Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Multiply and collect terms.
Factor.
Solve.
x2 + x – 12 = 0
log12
x + log12
(x +1) = 1
(x – 3)(x + 4) = 0
x – 3 = 0 or x + 4 = 0 Set each of the factors equal to zero.
x = 3 or x = –4
log12
x + log12
(x +1) = 1
log12
3 + log12
(3 + 1) 1log
123 + log
124 1
log12
12 1
The solution is x = 3.1 1
log12
( –4) + log12
(–4 +1) 1
log12
( –4) is undefined.
x
Check Check both solutions in the original equation.
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
Use 10 as the base for both sides.Use inverse properties on the right side.
Product Property of Logarithms.
Power Property of Logarithms.
Holt Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve.
2log x – log 4 = 0
Write as a quotient.
x = 2
Use 10 as the base for both sides.
Use inverse properties on the left side.
2log( ) = 0 x 4
2(10log ) = 100
x 4
2( ) = 1 x 4
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