Lesson 5.15

Solving Exponential Equations (Same Base)

When both sides of an equation can be written with the same base, you can set the exponents equal and solve — no logarithms needed.

Introduction

The key principle: if , then (when ). This is the fastest way to solve exponential equations when it applies.

Past Knowledge

Exponent rules (Unit 1), exponential functions (5.1–5.2).

Today's Goal

Solve equations by rewriting both sides with the same base.

Future Success

When same-base isn't possible, 5.16 uses logarithms instead.

Key Concepts

One-to-One Property

Same base → exponents must be equal

Strategy

1. Rewrite each side as a power of the same base

2. Set the exponents equal

3. Solve the resulting equation

Worked Examples

Example 1: Powers of 2

Basic

Solve .

1

Rewrite 16 as a power of 2

2

Set exponents equal

Example 2: Both Sides Need Rewriting

Intermediate

Solve .

1

Rewrite with base 3

2

Set exponents equal

Example 3: Negative Exponents

Advanced

Solve .

1

Rewrite with base 2

2

Set exponents equal and solve

Common Pitfalls

Power-of-a-Power Rule

, not . Multiply exponents when raising a power to a power.

Fractions as Negative Powers

, not . That's , a very different number.

Real-Life Applications

Same-base problems appear when quantities double, triple, or halve at regular intervals. For example, "bacteria triple every 2 hours; when will there be 243 times the original count?" Since 243 = 3⁵, this is a clean same-base problem.

Practice Quiz

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