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6-2: Properties of Logarithms Unit 6: Exponents/Logarithms English Casbarro You can find these properties because of exponential rules. a. bmbn= bm+n b. bm= bm-n bn c. (bm)n= bmn Example 1: a. log42 + log432 Express as a single logarithm. Simplify. b. log5625 + log525 c. log1 27 + log1 3 3 1 9 Example 2: a. log232 – log24 Express as a single logarithm. Simplify. b. log749 – log77 c. log216 – log22 Example 3: a. log3814 Express as a single logarithm. Simplify. 1 3 b. log5 ( 5 ) c. log5252 Exponential and Logarithmic are inverses, so They “undo” each other. For any base b, such that b > 0 and b ≠ 1 Algebra Example logbbx = x log10107 = 7 blogbx= x 10log102= 2 Change of base formula For a > 0 and a ≠ 1 and any base b, such that b > 0 and b ≠ 1 Algebra logb x = Note: Example loga x loga b log4 8 = log2 8 log2 4 This is most often used to change the base to 10 or e so that you can use your calculator. Example 4: a. log8 8 d. 3 x +1 log4 8 Simplify each expression. b. log 125 5 e. log9 27 c. log2 27 2 f. log8 16 Not all logarithms involve strictly numbers; some also involve variables. The properties work exactly the same way. Example 5: Write as a single logarithm. These properties are used to evaluate expressions as well. Example 6: a. c. b. d. Example 7: Solve the following problem for x using the properties. log3 ( x 2 + 7 x - 5) = log3 (6 x +1) Example 8: Solve the following problem for x by using the properties. 2loga 3 + loga ( x - 4) = loga ( x + 8)