The change-of-base formula

It is often necessary to pass from one base to another, for example to compute with a calculator a logarithm in a base other than $10$ or $e$.

Property — Change-of-base formula

For every admissible $a, b, x$: $\log_a x = \frac{\log_b x}{\log_b a}.$ From which, in particular, one obtains $\log_a x = \dfrac{1}{\log_x a}$ (case $b = x$): inversion of the base.

Example — Computing $\log_8 16$

With change of base $b=2$: $\log_8 16 = \frac{\log_2 16}{\log_2 8} = \frac{4}{3}.$

Remark — The "multiplied" version

The same formula can be rewritten as $\log_a x \cdot \log_b a = \log_b x$, that is, as a “product that carries” the logarithm from one base to another. It is worth being able to read it in both directions.

Links

Topics: Logarithmic function
Concepts: Change of base · Logarithm
Functions: Logarithmic function
Skills: Calculate · Use formulae