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ss_4_5

# ss_4_5 - Section 4.5 Logarithmic and Exponential Equations...

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Section 4.5 Logarithmic and Exponential Equations The logarithm and exponential functions are one-to-one functions and are inverse functions of each other. Hence If a x = a y , then x = y . If log a M = log a N , then M = N . a log a x = x log a a x = x The following examples illustrate the preceding rules. Example. Solve log 3 (3 x - 2) = 2. Solution. log 3 (3 x - 2) = 2 3 log 3 (3 x - 2) = 3 2 = 9 3 x - 2 = 9 x = 11 3 Example. Solve 3 log 2 x = - log 2 27. Solution. 3 log 2 x = - log 2 27 log 2 x 3 = log 2 27 - 1 x 3 = 27 - 1 = 1 27 x = 1 3 Example. Solve 5 1 - 2 x = 1 5 . Solution. 5 1 - 2 x = 1 5 = 5 - 1 implies 1 - 2 x = - 1 x = 1 Example. Solve 2 x +1 = 5 1 - 2 x . Solution. 2 x +1 = 5 1 - 2 x ln 2 x +1 = ln 5 1 - 2 x ( x + 1) ln 2 = (1 - 2 x ) ln 5 x (ln 2 + 2 ln 5) = ln 5 - ln 2 1

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x (ln 50) = ln 5 2 x = ln 5 2 ln 50 0 . 230 (Calculator) Example. Solve e x +3 = π x . Solution. e x +3 = π x ln e x +3 = ln π x ( x + 3) ln e = x ln π (ln e = 1) x + 3 = x ln π x (ln π - 1) = 3 x = 3 ln π - 1 20 . 728 (Calculator) Example. Solve log 2 8 x = 3. Solution. log 2 8 x = 3 log 2 (2 3 ) x = 3 log 2 2 3 x = 3 3 x = 3 x = 1 Example. Solve log 2 (3 x + 2) - log 4 x = 3. Solution. log 2 (3 x + 2) - log 4 x = 3 log 2 (3 x + 2) - log 2 x log 2 4 = 3 (Change of base formula) 2 log 2 (3 x +2) - log 2 x 2 = 2 3 = 8 (log 2 4 = 2) 2 log 2 (3 x +2) · 2 - log 2 x = 8 (2 a - b = 2 a 2 - b ) (3 x + 2) · 1 x = 8 3 x
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ss_4_5 - Section 4.5 Logarithmic and Exponential Equations...

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