Lesson 5.6

Intro to Logarithms

A logarithm answers the question: "What exponent gives me this number?" If , then . The log is the exponent.

Introduction

You know how to solve (take the square root). But how do you solve ? You need the inverse of exponentiation — that's what a logarithm is.

Past Knowledge

Exponential functions (5.1–5.2), inverse operations.

Today's Goal

Understand the definition of a logarithm and read log notation.

Future Success

5.7 covers converting between forms; Chapter 18 uses log properties to solve equations.

Key Concepts

The Definition

"log base of equals " means " to the equals "

Think of it as solving for the exponent:

"2 to what power gives 8?" → , so the answer is 3.

Key Facts

because

log "undoes" the exponent

exponent "undoes" the log

Restrictions:

Worked Examples

Example 1: Evaluate by Definition

Basic

Evaluate .

Ask: "3 to what power = 81?"

Example 2: Fractional Results

Intermediate

Evaluate .

Ask: "4 to what power = 1/16?"

Since , we need

Example 3: Log Equals 0 or 1

Quick

Common Pitfalls

Log of a Negative

is undefined. No power of a positive base can produce a negative number. The input to a log must be positive.

Log Is Not Multiplication

does not mean . The subscript is the base, and the number after is the argument.

Real-Life Applications

The Richter scale, decibel scale, and pH scale are all logarithmic. An earthquake measuring 7 is 10 times stronger than one measuring 6 — that's the log in action.