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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Exponential and Logarithmic Functions9
Exponential and Logarithmic Functions
9.1 Composite and Inverse Functions
9.2 Exponential Functions
9.3 Logarithmic Functions
9.4 Properties of Logarithmic Functions
9.5 Common and Natural Logarithms
9.6 Solving Exponential and Logarithmic Equations
9.7 Applications of Exponential and Logarithmic Functions
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3. Solving Exponential and Logarithmic Equations
9.6
Solving Exponential and Logarithmic Equations
Solving Exponential Equations
Solving Logarithmic Equations
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4. Solving Exponential Equations
Equations with variables in exponents, such as 3x = 5 and 73x = 90 are called exponential equations. In Section 9.3, we solved certain exponential equations by using the principle of exponential equality.
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5. The Principle of Exponential Equality
For any real number b, where
(Powers of the same base are equal if and only if the exponents are equal.)
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6. Example Solution Solve: 5x = 125.
Note that 125 = 53. Thus we can write each side as a power of the same base:
5x = 53
Since the base is the same, 5, the exponents must be equal. Thus, x must be 3. The solution is 3.
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7. The Principle of Logarithmic Equality
For any logarithmic base a, and for x, y > 0,
(Two expressions are equal if and only if the logarithms of those expressions are equal.)
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8. Example Solution Solve: 3 x +1 = 43 We have 3 x +1 = 43
Principle of logarithmic equality
log 3 x +1 = log 43
(x +1)log 3 = log 43
Power rule for logs
x +1 = log 43/log 3
x = (log 43/log 3) – 1
The solution is (log 43/log 3) – 1, or approximately
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9. Example Solution Solve: e1.32t = 2000 We have: e1.32t = 2000
Note that we use the natural logarithm
ln e1.32t = ln 2000
Logarithmic and exponential functions are inverses of each other
1.32t = ln 2000
t = (ln 2000)/1.32
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10. To Solve an Equation of the Form at = b for t
Take the logarithm (either natural or common) of both sides.
Use the power rule for exponents so that the variable is no longer written as an exponent.
Divide both sides by the coefficient of the variable to isolate the variable.
If appropriate, use a calculator to find an approximate solution in decimal form.
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11. Solving Logarithmic Equations
Equations containing logarithmic expressions are called logarithmic equations. We saw in Section 9.3 that certain logarithmic equations can be solved by writing an equivalent exponential equation.
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12. Example Solution Solve: log2(6x + 5) = 4. log2(6x + 5) = 4 6x + 5 = 24
The solution is 11/6. The check is left to the student.
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13. Often the properties for logarithms are needed
Often the properties for logarithms are needed. The goal is to first write an equivalent equation in which the variable appears in just one logarithmic expression. We then isolate that expression and solve as in the previous example.
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14. Example Solution Solve: log x + log (x + 9) = 1.
log[x(x + 9)] = 1
x(x + 9) = 101
x2 + 9x = 10
x2 + 9x – 10 = 0
(x – 1)(x + 10) = 0
x – 1 = 0 or x + 10 = 0
x = 1 or x = –10
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15. Check x = 1: log 1 + log (1 + 9) 0 + log (10) 0 + 1 = 1 x = –10:
= 1
TRUE
x = –10:
log (–10) + log (–10 + 9)
FALSE
The logarithm of a negative number is undefined. The solution is 1.
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16. Example Solution Solve: log3(2x + 3) – log3(x – 1) = 2.
log3[(2x + 3)/(x – 1)] = 2
(2x + 3)/(x – 1) = 32
(2x + 3)/(x – 1) = 9
(2x + 3) = 9(x – 1)
2x + 3 = 9x – 9
x = 12/7
The solution is 12/7. The check is left to the student.
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