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Published byAudrey Keating Modified over 3 years ago

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Warm-Up Solve each equation for x. Round your answers to the nearest hundredth. 4 minutes 1) 10 x = ) 10 x = Find the value of x in each equation. 3) x = log 4 1 4) ½ = log 9 x

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6.4.1 Properties of Logarithmic Functions Properties of Logarithmic Functions Objectives: Simplify and evaluate expressions involving logarithms Solve equations involving logarithms

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Properties of Logarithms Product Property log b (mn) = log b m + log b n For m > 0, n > 0, b > 0, and b 1:

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Example 1 = log log 5 10 log = given: log log log 5 (12)(10)

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Properties of Logarithms Quotient Property For m > 0, n > 0, b > 0, and b 1: log b = log b m – log b n m n

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Example 2 given: log log log = log 5 12 – log – = log

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Properties of Logarithms Power Property For m > 0, n > 0, b > 0, and any real number p: log b m p = p log b m

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Example 3 given: log log log = 4 3 = 12 = 4 log x = = 125 x = 3

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Practice Write each expression as a single logarithm. 1) log 2 14 – log 2 7 2) log 3 x + log 3 4 – log 3 2 3) 7 log 3 y – 4 log 3 x

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Homework p.382 #13-21 odds,31,35

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Warm-Up Write each expression as a single logarithm. Then simplify, if possible. 4 minutes 1) log log 6 30 – log 6 5 2) log 6 5x + 3(log 6 x – log 6 y)

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6.4.2 Properties of Logarithmic Functions Properties of Logarithmic Functions Objectives: Simplify and evaluate expressions involving logarithms Solve equations involving logarithms

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Properties of Logarithms Exponential-Logarithmic Inverse Property log b b x = x For b > 0 and b 1: and b log b x = x for x > 0

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Example 1 Evaluate each expression. a) b)

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Practice Evaluate each expression. 1) 7 log 7 11 – log ) log log 3 8

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Properties of Logarithms One-to-One Property of Logarithms If log b x = log b y, then x = y For b > 0 and b 1:

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Example 2 Solve log 2 (2x 2 + 8x – 11) = log 2 (2x + 9) for x. log 2 (2x 2 + 8x – 11) = log 2 (2x + 9) 2x 2 + 8x – 11 = 2x + 9 2x 2 + 6x – 20 = 0 2(x 2 + 3x – 10) = 0 2(x – 2)(x + 5) = 0 x = -5,2 Check:log 2 (2x 2 + 8x – 11) = log 2 (2x + 9) log 2 (–1) = log 2 (-1) undefined log 2 13 = log 2 13 true

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Practice Solve for x. 1) log 5 (3x 2 – 1) = log 5 2x 2) log b (x 2 – 2) + 2 log b 6 = log b 6x

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Homework p.382 #29,33,37,43,47,49,51,57,59,61

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