Lesson 5.7

Converting Forms

Every logarithmic statement has an equivalent exponential statement and vice versa. Fluency in switching between and is the key to solving both types of equations.

Introduction

Think of it like a translation: and say the exact same thing in two different notations. Being able to switch instantly is the most important skill in this chapter.

Past Knowledge

Definition of logarithm (5.6).

Today's Goal

Convert fluently between log and exponential form.

Future Success

This conversion is used in every subsequent lesson on solving equations.

Key Concepts

The Conversion Rule

Logarithmic Form

Exponential Form

Same three numbers, two arrangements: = base, = exponent, = result

Worked Examples

Example 1: Log → Exponential

Basic

Convert to exponential form.

Identify: base = 5, exponent = 3, result = 125

Example 2: Exponential → Log

Basic

Convert to logarithmic form.

Identify: base = 10, exponent = −2, result = 0.01

Example 3: Solving by Converting

Intermediate

Solve .

Convert to exponential form

Common Pitfalls

Swapping Base and Argument

In , the subscript is the base, is the argument. Don't swap them: , not .

Forgetting the Direction

The log output is the exponent. The argument is the result of exponentiation. Keep the roles straight!

Real-Life Applications

Converting between forms is how scientists move between "how long until...?" (solving for ) and "how much after...?" (evaluating the exponential). It's the bridge between the two perspectives.