Logarithmic Functions Section 4.2 Part 1 Logarithmic Functions Objectives: Change from logarithmic to exponential form. Change from exponential to logarithmic form. Evaluate logarithms. Use basic logarithmic properties.

Definition of a Logarithmic Function For x > 0 and b > 0, b = 1, y = logb x is equivalent to by = x. The function f (x) = logb x is the logarithmic function with base b.

Location of Base and Exponent in Exponential and Logarithmic Forms Logarithmic form: y = logb x Exponential Form: by = x. Exponent Exponent Base Base

Practice #1 Write each equation in its equivalent exponential form. a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y

With the fact that y = logb x means by = x, Solution With the fact that y = logb x means by = x, a. 2 = log5 x means 52 = x. Logarithms are exponents. b. 3 = logb 64 means b3 = 64. Logarithms are exponents. c. log3 7 = y or y = log3 7 means 3y = 7.

Practice #2 Evaluate a. log2 16 b. log3 9 c. log25 5 Solution

First, rewrite each expression as a logarithmic equation and convert Solution First, rewrite each expression as a logarithmic equation and convert to an exponential equation. Logarithmic Equation Exponential Equation Question Needed for Evaluation Evaluation a. log216 = x 2x = 16 2 to what power is 16? 4 b. log39 = x 3x = 9 3 to what power is 9? 2 c. log255 = x 25x = 5 25 to what power is 5? Since you must take the square root of 25 to get 5, that is the same as an exponent of ½. The answer is ½ .

Basic Logarithmic Properties Involving One logb b = 1 because 1 is the exponent to which b must be raised to obtain b. (b1 = b). logb 1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).

Inverse Properties of Logarithms For x > 0 and b  1, logb bx = x The logarithm with base b of b raised to a power equals that power. b logb x = x b raised to the logarithm with base b of a number equals that number.

Practice #3 a. log1111 b. log446 c.