8­5 Exponential and Logarithmic Equations
April 08, 2009
8-5 Exponential & Logarithmic
Equations
Objectives:
• Solve exponential equations.
• Solve logarithmic equations.
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Check Skills You'll Need
Evaluate each logarithm.
1. log981 log93
2. log 10 log39
3. log216 ÷ log28
4. Simplify 125
­ 23
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Solving Exponential Equations
An equation of the form bcx=a, where the exponent
includes a variable, is an exponential equation.
If m and n are positive and m = n, then log m = log n.
Therefore, you can solve an exponential equation by
taking the logarithm of each side of the equation.
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Example #1: Solving an Exponential Equation
Solve 73x = 20.
73x = 20
log 73x = log 20
3x log 7 = log 20
x=
log 20
3log 7
x ≈ 0.5132
Check:
Take the common logarithm of each side.
Use the power property of logarithms.
Divide each side by 3 log 7.
Use a calculator.
73x = 20
73(0.5132) = 20
20.00382 ≈ 20
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Example #2: Solve each equation. Round to the
nearest ten-thousandth. Check your answers.
a. 3x = 4
b. 62x = 21
c. 3x+4 = 101
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Solving Logarithmic Equations
To evaluate a logarithm with any base, you can use the
Change of Base Formula.
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Example #3: Using the Change of Base Formula
Use the Change of Base Formula to evaluate log315.
log315 =
log 15
log 3
≈ 2.4650
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Example #4: Evaluate log5400.
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8­5 Exponential and Logarithmic Equations
April 08, 2009
An equation that includes a logarithmic
expression, such as log315 = log2x is called a
logarithmic equation.
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Example #5: Solving a Logarithmic Equation
Solve log (3x + 1) = 5.
log (3x + 1) = 5
3x + 1 = 105
3x + 1 = 100,000
3x = 99,999
x = 33,333
Check:
log (3x + 1) = 5
log (3(33,333) + 1) = 5
log (100,000) = 5
log 105 = 5
5=5
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Example #6: Solve log (7 ­ 2x) = ­1. Check your answer.
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Example #7: Using Logarithmic Properties to Solve an Eqauation
Solve 2 log x ­ log 3 = 2.
2 log x ­ log 3 = 2
log ( ) = 2
x2
3
x2
3
= 102
Write as a single logarithm.
Write in exponential form.
x2 = 3(100)
x = ±10√3 ≈ ±17.32
Log x is defined only for x>0, so the solution is 10√3 or about 17.32.
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Example #8: Solve log 6 ­ log 3x = ­2.
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8­5 Exponential and Logarithmic Equations
April 08, 2009
Homework: page 464
(1 - 12, 23, 25 - 32 evaluate, 33 - 45)
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