8.3 Properties of logarithms ©2006 by R. Villar All Rights Reserved

Find the inverse of each: 1. g(x) = 5x 2. f(x) = 2 + log4x Warm-up Find the inverse of each: 1. g(x) = 5x 2. f(x) = 2 + log4x Simplify each: 3. 4. 5. x = 5y y = log5 x x = 2 + log4 y x – 2 = log4 y y = 4x – 2 x 7 x 15 x 6

67. 68. 69. 70. 59. g(x) = 6x 60. g(x) = log8x 61. g(x) = log1/3 x 62. g(x) = (1/2)x

Properties of Logarithms Consider the following two problems: Simplify log3 (9 • 27) = log3 (32• 33) = log3 (32 + 3) = 2 + 3 Simplify log3 9 + log3 27 = log3 32 + log3 33 = 2 + 3 These examples suggest the following property: Product Property of Logarithms: For all positive numbers m, n and b where b ≠ 1, logb mn = logb m + logb n

We will use the Product Property of Logarithms to solve problems... Example Given log2 5 = 2.322, find log2 40 log2 20 = log2 (23 • 5) = log2 23 + log2 5 = 3 + 2.322 = 5.322

Consider the following: a. b. = log3 34 33 = log3 34 – 3 = 4 – 3 = log3 34 – log3 33 = 4 – 3 These examples suggest the following property: Quotient Property of Logarithms: For all positive numbers m, n and b where b ≠ 1, logb m = logb m – logb n n

= log12 9 12 = log12 9 – log12 12 = 0.884 – 1 = –0.116 = log12 18 9 = log12 18 – log12 9 = 1.163 – 0.884 = 0.279 Examples: Given log12 9 = 0.884 and log12 18 = 1.163, find each: a. b. log12 2

Consider the following: Evaluate a. log3 94 b. 4 log3 9 = 2 • 4 = (log3 32) • 4 = 2 • 4 These examples suggest the following property: Power Property of Logarithms: For all positive numbers m, n and b where b ≠ 1, logb mp = p • logb m Consider the following: Evaluate a. log3 94 b. 4 log3 9

Example: Expand log10 7x3 log10 7 + log10 x3 log10 7 + 3log10 x Example: Expand log2 85/3x2y4 log2 85/3 + log2 x2 + log2 y4 5 log2 8 + 2log2 x + 4log2 y 3

Example: Expand loga 4xy2 z3 This logarithm contains several operations that can be expanded… Multiplication expands to addition; The exponent expands to multiplication; Division expands to subtraction… loga 4 + loga x + 2 loga y – 3 loga z

Assignment p. 416: 5 – 24 all