lecture 12 sections 4.5 logarithmic equations
TRANSCRIPT
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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