Name
LESSON
7-5
Date
Class
Reteach
Exponential and Logarithmic Equations and Inequalities
An exponential equation contains an expression that has a variable as an exponent.
5 x ⫽ 25 is an exponential equation.
x ⫽ 2, since 5 2 ⫽ 25.
If x ⫽ y, then
log x ⫽ log y
x ⬎ 0 and y ⬎ 0 .
Remember: You can take the logarithm of both sides of an exponential
equation. Then use other properties of logarithms to solve.
Solve 6 x ⫹ 2 ⫽ 500.
Step 1 Since the variable is in the exponent, take the log of both sides.
6 x ⫹ 2 ⫽ 500
x⫹2
⫽ log 500
log 6
Step 2
Use the Power Property of Logarithms: log a p ⫽ p log a.
x⫹2
⫽ log 500
log 6
x ⫹ 2 log 6 ⫽ log 500
“Bring down” the exponent to multiply.
Step 3
Isolate the variable. Divide both sides by log 6.
x ⫹ 2 log 6 ⫽ log 500
log 500
x ⫹ 2 ⫽ _______
log 6
Solve for x. Subtract 2 from both sides.
log 500
x ⫽ _______ ⫺ 2
log 6
Use a calculator to approximate x.
x 1.468
Step 4
Step 5
Step 6
Use a calculator to check.
6 1.468 ⫹ 2 499.607
Solve and check.
1. 4
⫺x
⫽ 32
log 4
⫺x
⫽ log 32
log 3
⫺x log 4 ⫽ log 32
4x
⫺2.5 ⫽ 32
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c07-5_rt.indd 38
⫽ log 90
4x log 3 ⫽ log 90
x ⫽ ⫺2.5
4⫺
3. 5 x ⫺ 3 ⫽ 600
2. 3 4x ⫽ 90
x 1.024
34
1.024 38
90.01
log 5
x
x⫺3
⫽ log 600
⫺ 3 log 5 ⫽ log 600
x 6.975
5 6.975 ⫺ 3 600.352
Holt Algebra 2
5/15/06 1:38:35 PM
Process Black
Name
LESSON
7-5
Date
Class
Reteach
Exponential and Logarithmic Equations and Inequalities
(continued)
A logarithmic equation contains a logarithmic expression that has a variable.
log 5 x ⫽ 2 is a logarithmic equation.
2
x ⫽ 25, since 5 ⫽ 25.
Combine and use properties of logarithms to solve logarithmic equations.
Solve: log 80x ⫺ log 4 ⫽ 1
Step 1 Use the Quotient Property of Logarithms.
log 80x ⫺ log 4 ⫽ 1
80x ⫽ 1
log ____
4
Step 2 Simplify.
80x ⫽ 1
log ____
4
log 20x ⫽ 1
Step 3 Use the definition of the logarithm:
x
if b ⫽ a, then log b a ⫽ x.
x
logx ⫺ logy ⫽ log __
y
Remember: Use 10 as the base
when the base is not given.
log 10 20x ⫽ 1
1
10 ⫽ 20x
Step 4 Solve for x. Divide both sides by 20.
10 ⫽ 20x
1⫽x
__
2
Solve and check.
4
4. log 3 x ⫽ 8
5. log 4 ⫹ log x ⫹ 2 ⫽ 2
4 log 3 x ⫽ 8
log 4 x ⫹ 2 ⫽ 2
8
log 3 x ⫽ __
4
log 10 4x ⫹ 8 ⫽ 2
4x ⫹ 8 ⫽ 10 2
6. log 75x ⫺ log 3 ⫽ 1
75x ⫽ 1
log ____
3
log 25x ⫽ 1
1
4x ⫹ 8 ⫽ 100
10 ⫽ 25x
32 ⫽ x
4x ⫽ 92
x⫽9
x ⫽ 23
10 ⫽ 25x
2
x ⫽ __
5
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c07-5_rt.indd 39
39
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