8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
Objectives:
• Solve exponential equations.
•
Solve logarithmic equations.
1

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
Check Skills You'll Need
Evaluate each logarithm.
1. log
9
81 log
9
3 2. log 10 log
3
9
3. log
2
16
log
2
8 4. Simplify 125
­ 2
3
2

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
3

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
Solving Exponential Equations
An equation of the form b cx =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.
4

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
Example #1: Solving an Exponential Equation
Solve 7 3x = 20.
7 3x = 20 log 7 3x = log 20
3x log 7 = log 20 x = log 20
3log 7 x ≈ 0.5132
Take the common logarithm of each side.
Use the power property of logarithms.
Divide each side by 3 log 7.
Use a calculator.
Check: 7 3x = 20
7 3(0.5132) = 20
20.00382 ≈ 20
5

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
Example #2: Solve each equation. Round to the nearest ten-thousandth. Check your answers.
a. 3 x = 4 b. 6 2x = 21 c. 3 x+4 = 101
6

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
Solving Logarithmic Equations
To evaluate a logarithm with any base, you can use the
Change of Base Formula.
7

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
Example #3: Using the Change of Base Formula
Use the Change of Base Formula to evaluate log
3
15.
log
3
15 = log 15 log 3
≈ 2.4650
8

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
Example #4: Evaluate log
5
400.
9

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
An equation that includes a logarithmic expression, such as log
3
15 = log
2 x is called a logarithmic equation.
10

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
Example #5: Solving a Logarithmic Equation
Solve log (3x + 1) = 5.
log (3x + 1) = 5
3x + 1 = 10 5
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 10 5 = 5
5 = 5
11

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
Example #6: Solve log (7 ­ 2x) = ­ 1.
Check your answer.
12

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
Example #7: Using Logarithmic Properties to Solve an Eqauation
Solve 2 log x
log 3 = 2.
2 log x
log 3 = 2
3
2
log = 2
Write as a single logarithm.
x 2
= 10 2
Write in exponential form.
3
x 2 = 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.
13

8­5 Exponential and Logarithmic Equations 2011
Example #8: Solve log 6 ­ log 3x = ­ 2.
April 29, 2011
14

8­5 Exponential and Logarithmic Equations 2011 April 29, 2011
(1 - 12, 23, 25 - 32 evaluate, 33 - 45)
15