Section 8.4 Logarithmic Functions Evaluate logarithmic functions Graph logarithmic functions

Tell me what you know… 2 2 = 4 2 x = = 8

Definition of a logarithm with base b. Let b and y be positive numbers, b ≠ 1. The logarithm of y with base b is denoted by log b y and is defined as follows: log b y = x if and only if b x = y log b y is read as “log base b of y”

Logarithmic Form vs. Exponential Form log b y = x b x = y equivalent Given an equation in one of these forms, you can always rewrite it in the other form.

Example 1: Rewriting Log Equations Logarithmic Form a. log 2 32 = 5 b. log 5 1 = 0 c. log = 1 d. log = -1 e. log 1/2 2 = -1 Exponential Form

Special Log Values Let b be a positive real number such that b ≠ 1 LOGARITHM OF 1 LOGARITHM OF 1log b 1 = 0 because b 0 = 1 LOGARITHM OF BASE b LOGARITHM OF BASE blog b b = 1 because b 1 = b

Example 2: Evaluating Log Expressions without a calculator Evaluate the expression: a. log 3 81 = 3 to what power gives 81? b. log = 5 to what power gives 0.4?

Example 2: Evaluating Log Expressions without a calculator Evaluate the expression: c.log 1/2 8 = 1/2 to what power gives 8? d.log 9 3 = 9 to what power gives 3?

You try… Evaluate the log expression a. log 4 64 b. log c. log 1/4 256 d. log 32 2

Common Log vs. Natural Log Common Log log 10 x = log x Natural Log log e x = ln x

Example 3: Evaluating Common and Natural Logs Expression a. log 5 = b. ln 0.1 = c. log 7 = d. ln 0.25 =

EXAMPLE 4 Evaluate a logarithmic model Tornadoes The wind speed s (in miles per hour) near the center of a tornado can be modeled by where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado’s center. 93 log d + 65 s =

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