Introduction
Trapped with a calculator that only speaks "Log" and "Ln"? The Change of Base formula is your universal translator for any base you encounter.
Past Knowledge
You need to be comfortable using the or button on your calculator and evaluating fractions.
Today's Goal
We introduce the formula . This allows us to compute any logarithm using standard buttons.
Future Success
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
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.
Practice Quiz
Practice Quiz
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