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**Objectives Solve exponential and logarithmic equations and equalities.**

Solve problems involving exponential and logarithmic equations.

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Vocabulary exponential equation logarithmic equation

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An exponential equation is an equation containing one or more expressions that have a variable as an exponent. To solve exponential equations: Try writing them so that the bases are all the same. Take the logarithm of both sides.

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**When you use a rounded number in a check, the result will not be exact, but it should be reasonable.**

Helpful Hint

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Check It Out! Example 1a Solve and check. 32x = 27

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Check It Out! Example 1b Solve and check. 7–x = 21

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Check It Out! Example 1c Solve and check. 23x = 15

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Check It Out! Example 2 You receive one penny on the first day, and then triple that (3 cents) on the second day, and so on for a month. On what day would you receive a least a million dollars.

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A logarithmic equation is an equation with a logarithmic expression that contains a variable. You can solve logarithmic equations by using the properties of logarithms.

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**Review the properties of logarithms from Lesson 7-4.**

Remember!

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Check It Out! Example 3a Solve. 3 = log 8 + 3log x

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Check It Out! Example 3b Solve. 2log x – log 4 = 0

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**Watch out for calculated solutions that are not solutions of the original equation.**

Caution

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Check It Out! Example 4a Use a table and graph to solve 2x = 4x – 1. Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) as Y2. In the table, find the x-values where Y1 is equal to Y2. In the graph, find the x-value at the point of intersection. The solution is x = 2.

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Check It Out! Example 4b Use a table and graph to solve 2x > 4x – 1. Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) as Y2. In the table, find the x-values where Y1 is greater than Y2. In the graph, find the x-value at the point of intersection. The solution is x < 2.

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Check It Out! Example 4c Use a table and graph to solve log x2 = 6. Use a graphing calculator. Enter log(x2) as Y1 and 6 as Y2. In the table, find the x-values where Y1 is equal to Y2. In the graph, find the x-value at the point of intersection. The solution is x = 1000.

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