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Math 152: Peacemaker Blitzer/9.4 Properties of Logarithms Let b , m , and n be positive real numbers with b ≠ 1 Rule 1. log b (mn) = logb m + log b n 2. m log b = log b m − log b n n 3. log b m p = p log b m Example 1. Using Logarithmic Properties to Expand Logarithmic Expressions Example 1 Use the properties of logarithms to expand each expression and evaluate if possible. a. log 5 25x c. log 9 e. ln x 2 y 3 g. 4 log8 ( x − y ) z 6 x 9 b. ln(4 x) d. log 3 ( x + 3) f. 3x log 5 y 2 2. Using Logarithmic Properties to Condense Logarithmic Expressions Example 2 Use the properties of logarithms to condense each expression. a. log 3 486 − log3 18 b. log 4 8 + log 4 32 c. ln x − 4 ln y d. 1 8 log b y + log b z 4 e. 1 3log 5 x + log 5 y − 4 log 5 z 2 3. Change of Base Property For any logarithmic bases a and b , and a real number M > 0 , log b M = log a M , so for practical purposes we log a b use the common logarithm or the natural logarithm: log b M = Example 3 a. log8 35 b. log16 5 log M log b log b M = ln M ln b Use the change of base formula to evaluate the logarithms. Math 152 – Blitzer/9.4 2