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**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.

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**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.

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**Location of Base and Exponent in Exponential and Logarithmic Forms**

Logarithmic form: y = logb x Exponential Form: by = x. Exponent Exponent Base Base

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**Practice #1 Write each equation in its equivalent exponential form.**

a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y

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**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.

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

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**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 .

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**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).

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**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.

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Practice #3 a. log1111 b. log446 c.

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