Lesson 8.6

Change of Base Formula

Trapped with a calculator that only speaks "Log" and "Ln"? The Change of Base formula is your universal translator for any base you encounter.

Introduction

We know that . But what about ? It's somewhere between 3 and 4. Most simple calculators don't have a button for base 2. We need a way to convert it to base 10 or base .

Prerequisite Connection

You need to be comfortable using the or button on your calculator and evaluating fractions.

Today's Increment

We introduce the formula . This allows us to compute any logarithm using standard buttons.

Why This Matters

In Computer Science, we often need Base 2 logs (binary search depth). In physics, we need Base 10 (decibels). In Calculus, we need Base . This formula lets us effortlessly switch between these worlds without re-deriving math.

Key Concepts

The Formula

To change from base to a new base :

Usually, we choose (natural log) for simplicity: .

This makes sense conceptually: we are asking "how many powers of fit into ?" which is the ratio of their sizes on a log scale.

Worked Examples

Example 1: Numerical Calculation

Calculate to 3 decimal places.

Apply Formula

Compute

Example 2: Verification

Show that using the Change of Base formula with base 2.

Set up Ratio

Evaluate Numerator/Denominator

(since ).
(since ).

Divide

Example 3: Graph Transformation (Advanced)

Rewrite in terms of to find the vertical scaling factor.

Apply Formula

Interpret Coefficient

The graph of

is just

vertically compressed by a factor of

Common Pitfalls

Base on Top?

Students often forget which log goes in the numerator.
Remember: The "base" goes in the "basement" (denominator).

Real-World Application

Astronomy (Magnitude)

The brightness of stars is measured on a logarithmic scale with base . To compare star brightness using modern data software (which uses natural logs), astronomers constantly use the Change of Base formula to normalize definitions of luminosity.