MATH 107

Section 4.5

Logarithmic and

Exponential Equations

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EXAMPLE 1 Solving an Exponential Equation

Solve each equation.

Solution

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OBJECTIVE Solve exponential equations when both sides are not expressed with the same base.

Step 1 Isolate the exponential expression on one side of the equation.

Step 2 Take the common or natural logarithm of both sides.

Step 3 Use the power rule, loga M r = r loga M.

EXAMPLE 2Solving Exponential Equations Using the Logarithms

EXAMPLE Solve for x: 5 ∙ 2x – 3 = 17.

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OBJECTIVE Solve exponential equations when both sides are not expressed with the same base.

Step 4 Solve for the variable.

EXAMPLE 2Solving Exponential Equations Using the Logarithms

EXAMPLE Solve for x: 5 ∙ 2x – 3 = 17.

Practice Problem

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6© 2010 Pearson Education, Inc. All rights reserved

EXAMPLE 3Solving an Exponential Equation with Different Bases

2 3 1ln5 ln3

ln5 ln3

2 ln5 3ln5 ln3 ln3

2 ln5 ln3 ln3 3ln5

2ln5 ln3 ln3 3ln5

ln3 3ln52.795

2ln5

3

n3

2 1

l

x x

x x

x x

x

x

x x

Solve the equation 52x–3 = 3x+1 and approximate the answer to three decimal places.

When different bases are involved, begin with Step 2.

Solution

Practice Problem

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8© 2010 Pearson Education, Inc. All rights reserved

SOLVING LOGARITHMS EQUATIONS

42log means 2 64 1x x

Equations that contain terms of the form log a x are called logarithmic equations.

To solve a logarithmic equation we write it in the equivalent exponential form.

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EXAMPLE 7 Solving a Logarithmic Equation

Solve:

We must check our solution.

Solution

2

2

2

1

4 3log 1

3log 1 4 3

log 1

2

1

2

x

x

x

x

x

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EXAMPLE 7 Solving a Logarithmic Equation

Solution continued

The solution set is

Check x =?

2

?

2

?

2

?

?

1

4 3log 1

14 3log 1

2

4 3log 1

4 3

2

1

1 1

x

Practice Problem

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12© 2010 Pearson Education, Inc. All rights reserved

EXAMPLE 9 Using the Product and Quotient Rules

Solve: 2 2a. log 3 log 4 1x x

2 2b. log 4 log 3 1.x x Solution

2 2

2

1

2

2

a. log 3 log 4 1

log 3 4 1

3 4 2

7 12 2

7 10 0

2 5 0

x x

x x

x x

x x

x x

x x

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EXAMPLE 9 Using the Product and Quotient Rules

Solution continued

Check x = 2

Logarithms are not defined for negative numbers, so x = 2 is not a solution.

2 2

2 2

log 3 log 4 1

log 1 lo

2 2

g 2 1

?

?

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EXAMPLE 9 Using the Product and Quotient Rules

Solution continued

The solution set is {5}.

Check x = 5

2 2

2 2

log 3 log 4 1

log 2 log 1 1

1 0 1

1 1

5 5

?

?

?

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EXAMPLE 9 Using the Product and Quotient Rules

Solution continued

b.

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EXAMPLE 9 Using the Product and Quotient Rules

Solution continuedCheck x = 2

1 1Check x = 5

log2 (1) and log2 (2) are undefined, so solution set is { 2}.

Practice Problem

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