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8 – 4 : Logarithmic Functions (Day 1) Objective: Be able to evaluate Logarithmic Functions. You know that 2 4 and 2 8. However, 2 3 for what value of x does 2 6? It is reasonalbe x to assume that x must lie between 2 and 3. To find the exact value of x, mathematicans defined logarithms. Definition of Logarithm with Base b Let b and y be positive numbers b ≠ 1. The logarithm of y base b is denoted by Logby and is defined as follows: logb y x if and only if b y x The expression logb y is read "log base b of y." log b y x and b y x logarithmic form exponential form Example 1: Rewriting Logarithmic Expressions A.) log232 = 5 25 = 32 B.) log51 = 0 50 = 1 C.) log101 = 1 101 = 10 D.) log10 0.1 = -1 10-1 = 0.1 E.) log1/2 2 = -1 (1/2)-1 = 2 Special Logarithm Values Let b be a positive real number such that b ≠ 1 Logarithm of 1 logb1 0 because b 1 0 Logarithm of Base b logb b 1 because b b 1 Example 2: Evaluating Logarithmic Expressions Evaluate the expression. a. log3 81 x 3x = 81 3 raised to what power equals 81? 3 81 log 3 81 4 4 b. log 5 0.04 x 5x = 0.04 5 raised to what power equals 0.04? 1 1 5 2 0.04 5 25 log5 0.04 2 2 c. log 1 8 2 ½ raised to what power equals 8? 3 1 8 log 1 8 3 2 2 d. log 9 3 9 raised to what power equals 3? 1 2 1 9 3 log 9 3 2 Common Logarithm Natural Logarithm log10 x log x log e x ln x ln and e are inverses of each other Example 3: Evaluating Common and Natural Logarithms Expression a. log 5 b. ln 0.1 Keystrokes Display LOG 5 ENTER 0.69870 LN 0.1 -2.302585 Homework: P. 490 #16 – 34 even, 36 -47 all Example 4: Evaluating a Logarithmic Function The slope s of a beach is related to the average diameter d (in millimeters) of the sand particles on the beach by this equation. s 0.159 0.118log d Find the slope of a beach if the average diameter of the sand particles is 0.25 millimeters. Solution If d = 0.25, then the slope of the beach is: s 0.159 0.118log d 0.159 0.118log 0.25 0.159 0.118 0.602 0.159 0.071 0.09