Lesson 8.5

The Power Rule of Logarithms

The most powerful property of logarithms is their ability to bring an exponent down to the ground level, turning powers into multipliers.

Introduction

In Lesson 8.4, we saw that , which simplifies to . This pattern holds for any power, even non-integers.

Prerequisite Connection

Remember that . This multiplicative property of exponents directly creates the multiplicative property of logs.

Today's Increment

We formalize this as . The exponent hops over the log function to become a coefficient.

Why This Matters

This is the only tool we have to solve equations where the variable is stuck in the exponent, like . Without the Power Rule, we cannot solve for time variables in growth equations.

Key Concepts

The Power Rule

The logarithm of a power is the exponent times the logarithm of the base.

Since radicals are fractional exponents (), this rule also handles roots: .

Worked Examples

Example 1: Basic Expansion

Expand .

Bring down the exponent

Example 2: Condensing with Coefficients

Write as a single logarithm: .

Apply Power Rule in Reverse

Move coefficients to exponents: .

Apply Product Rule

Example 3: Complex Deconstruction

Expand completely: .

Outer Root to Power

. Bring down the 1/2: .

Quotient and Power

Inside the brackets: .

Distribute

Common Pitfalls

Power of the Log vs. Log of the Power

. In the first case, you compute the log then square the result. In the second, you square x then take the log. The Power Rule ONLY applies if the exponent is inside the argument.

Real-World Application

Carbon Dating

The formula for radioactive decay is . To solve for the age of a fossil (), scientists must isolate the exponent. Taking the natural log of both sides allows the to come down, making the calculation possible.