Lesson 5.13

Power Property of Logarithms

The log of a power lets you bring the exponent down as a coefficient. This is the property that makes it possible to solve for exponents — the core skill of Chapter 18.

Introduction

If lets the exponent "come down front," then we can solve equations like by taking a log and bringing down where we can isolate it. This is the key that unlocks exponential equations.

Past Knowledge

Product (5.11), quotient (5.12), power rule for exponents.

Today's Goal

Use the power property to move exponents in and out of logs.

Future Success

Essential for solving exponential equations in 5.15–5.16.

Key Concepts

Power Property

The exponent "comes down" as a multiplier in front of the log

All Three Properties

Product

Quotient

Power

Worked Examples

Example 1: Bringing Exponent Down

Basic

Simplify .

Verify: ✓

Example 2: Roots as Fractional Powers

Intermediate

Expand .

Roots become fractional coefficients!

Example 3: Moving Coefficient Up

Reverse

Rewrite as a single log.

The coefficient becomes the exponent — reversing the power property.

Common Pitfalls

Coefficient vs. Exponent on the Log Itself

. The power property moves the exponent on the argument, not on the log expression itself.

Forgetting Roots Are Powers

. Always convert roots to fractional exponents before applying the power property.

Real-Life Applications

The power property is how we "unwrap" an unknown exponent. Every real-world problem asking "how long until…?" in exponential growth — doubling time, half-life, time to reach a target balance — is solved by bringing the exponent down with this property.