5.3 Intro to Logarithms 2/27/2013

Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic functions are the inverse of exponential functions Example: log 2 8 = 3 since 2 3 = 8 Read as: “log base 2 of 8”

Location of Base and Exponent in Exponential and Logarithmic Forms Logarithmic form: x = log b y Exponential Form: b x = y Exponent Base

Basic Logarithmic Properties Involving One Log b b = __ because 1 is the exponent to which b must be raised to obtain b. (b 1 = b). Log b 1 = __ because 0 is the exponent to which b must be raised to obtain 1. (b 0 = 1). 1 0 log b y = x if and only if b x = y

Popular Bases have special names Base 10 log 10 x = log x is called a common logarithm Base “e” log e x = ln x is called the natural logarithm or “natural log”

e and Natural Logarithm

Inverse properties

Example 1 Rewrite in Exponential Form LOGARITHMIC FORM a. log 2 16 = EXPONENTIAL FORM = 16 b. log 7 1 = = 1 c. log 5 5 = = 5 d. log 0.01 = 2 – = – e. log 1/4 4 = 1 – = – 1 log b y = x is b x = y

Example 1 Rewrite in Exponential Form LOGARITHMIC FORM EXPONENTIAL FORM log e x = ln x

Example 2 Rewrite in Logarithmic Form Form LOGARITHMIC FORM EXPONENTIAL FORM log b y = x is b x = y

Example 3 Evaluate Logarithmic Expressions Evaluate the expression. a. log ?4? = 64 What power of 4 gives 64 ? 4343 = 64 Guess, check, and revise. log 4 64 = 3 log b y = x is b x = y 4 1/2 = 2 Guess, check, and revise. log 4 2 = 2 1 4?4? = 2 What power of 4 gives 2 ? b. log 4 2

Example 3 Evaluate Logarithmic Expressions = – 2 Guess, check, and revise. = What power of gives 9 ? ? 3 1 log 1/3 9 = 2 – c. log 1/3 9

Example 4 Simplifying Exponential Functions a. b.

Example 4 Simplifying Exponential Functions c. d.

Homework WS 5.3 odd problems only