Change-of-Base Formula

Why a change-of-base formula is needed

Sometimes a logarithm needs to be evaluated in a base for which the value isn't directly known or easy to compute (e.g. $\log_4 64$), while an equivalent calculation in a DIFFERENT, more convenient base (e.g. base $2$) IS easy. The change-of-base formula converts a logarithm from one base to another:

Here $c$ can be ANY valid base — commonly $10$ (common logarithm) or a base that makes both the numerator and denominator easy to evaluate directly.

Worked reasoning

To evaluate $\log_4 64$, change to base $2$ (since both $4$ and $64$ are clean powers of $2$):

Check directly from the definition: $4^3 = 64$ ✓, confirming $\log_4 64=3$ without needing the change-of-base formula at all — a useful way to verify any change-of-base computation.