Section 4.4 Exponential and Logarithmic Equations
*Exponential Equations
- Solving Exponential Equation by Expressing Each Side as a Power of Same Base
If b M  b N , M  N .
1. Rewrite the equation in the form b M  b N .
2. Set M  N .
3. Solve for the variable. 
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Example 1) Solve:
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a. 5 3x6  125
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b. 8 x 2  4 x3 .
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- Using Natural Logarithm to Solve Exponential Equations
1. Isolate the exponential expression.
2. Take the natural logarithm on both sides of the equation.
3. Simplify using one of the following properties:
ln b x  x ln b or ln e x  x .
4. Solve for the variable.
Example 2) Solve: 5 x  134
set and then use a calculator to obtain a decimal
 . Find the solution
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approximation to two decimal places for the solution.
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Example 3) Solve: 7e 2x  5  58 . Find the solution set and then use a calculator to obtain a
decimal approximation to two decimal places for the solution.
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Example 4) Solve: 32x1  7 x 1. Find the solution set and then use a calculator to obtain a
decimal approximation to two decimal places for the solution.
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Example 5) Solve: e 2x  8e x  7  0 . Find the solution set and then use a calculator to obtain a
decimal approximation to two decimal places, if necessary, for the solution.
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*Logarithmic Equations
- Using the Definition of a Logarithm to Solve Logarithmic Equations
1. Express the equation in the form log b M  c .
2. Use the definition of a logarithm to rewrite the equation in exponential form:
log b M  c means b c  M .
3. Solve for the variable.
 in the original equation. Include in the solution set only values
4. Check proposed solutions
for which M  0 .
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Example 6) Solve:
a. log 2 (x  4)  3
b. 4 ln( 3x)  8 .
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Example 7) Solve: log x  log( x  3) 1.
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- Using the One-to-One Property of Logarithms to Solve Logarithmic Equations
1. Express the equation in the form log b M  log b N . This form involves a single logarithm
whose coefficient is 1 on each side of the equation.
2. Use the one-to-one property to rewrite the equation without logarithms:
If log b M  log b N , then M  N .
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3. Solve for the variable.
4. Check proposed solutions in the original equation. Include in the solution set only values
for which M  0 and N  0.
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Example 8) Solve: ln( x  3)  ln( 7x  23)  ln( x  1) .
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*Applications
Example 9) Medical research indicates that the risk of having a car accident increases
exponentially as the concentration of alcohol in the blood increases. The risk is modeled by
R  6e12.77x ,
where x is the blood alcohol concentration and R , given as a percent, is the risk having a car
accident. What blood alcohol concentration corresponds to a 7% risk of a car accident?
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Example 10) How long to the nearest tenth of a year, will it take $1000 to grow to $3600 at 8%
annual interest compounded quarterly?
Example 11) The function f (x)  34.1ln x 117.7 models the number of U.S. Internet users,
f (x) , in millions, x years after 1999. Find in which year there will be 210 million Internet
users in the United States.
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