2. CHAPTER 5: Exponential and Logarithmic Functions
5.1 Inverse Functions
5.2 Exponential Functions and Graphs
5.3 Logarithmic Functions and Graphs
5.4 Properties of Logarithmic Functions
5.5 Solving Exponential and Logarithmic Equations
5.6 Applications and Models: Growth and Decay; and Compound Interest
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3. 5.5 Solving Exponential and Logarithmic Equations
Solve exponential equations.
Solve logarithmic equations.
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4. Solving Exponential Equations
Equations with variables in the exponents, such as
3x = and x = 64,
are called exponential equations.
Use the following property to solve exponential equations.
Base-Exponent Property
For any a > 0, a  1,
ax = ay  x = y.
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5. Copyright © 2009 Pearson Education, Inc.
Example
Solve
Solution:
Write each side as a power of the same number (base).
Since the bases are the same number, 2, we can use the base-exponent property and set the exponents equal:
Check x = 4:
TRUE
The solution is 4.
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Another Property
Property of Logarithmic Equality
For any M > 0, N > 0, a > 0, and a  1,
loga M = loga N  M = N.
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Example
Solve: 3x = 20.
Solution:
This is an exact answer. We cannot simplify further, but we can approximate using a calculator.
We can check by finding  20.
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8. Copyright © 2009 Pearson Education, Inc.
Example
Solve: e0.08t = 2500.
Solution:
The solution is about 97.8.
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9. Solving Logarithmic Equations
Equations containing variables in logarithmic expressions, such as
log2 x = and log x + log (x + 3) = 1,
are called logarithmic equations.
To solve logarithmic equations algebraically, we first try to obtain a single logarithmic expression on one side and then write an equivalent exponential equation.
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10. Copyright © 2009 Pearson Education, Inc.
Example
Solve: log3 x = 2.
Check:
Solution:
TRUE
The solution is
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11. Copyright © 2009 Pearson Education, Inc.
Example
Solve:
Solution:
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12. Copyright © 2009 Pearson Education, Inc.
Example (continued)
Check x = 2:
Check x = –5:
FALSE
TRUE
The number –5 is not a solution because negative numbers do not have real number logarithms. The solution is 2.
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13. Copyright © 2009 Pearson Education, Inc.
Example
Solve:
Solution:
Only the value 2 checks and it is the only solution.
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14. Example - Using the Graphing Calculator
Solve: e0.5x – 7.3 = 2.08x
Solve:
Graph y1 = e0.5x – 7.3 and y2 = 2.08x and use the Intersect method.
The approximate solutions are –6.471 and
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