Log Change of Base

_b(x) = _c(x)/ _c(b) — switch to a base where you can see the answer.

The idea

Theorem (Change of base).

For positive numbers $b$, $c$ and $x$ with $b \ne 1$ and $c \ne 1$, $\log_b x = \frac{\log_c x}{\log_c b}.$

The change-of-base formula rewrites a logarithm in one base as a quotient of logarithms in any other. To derive it, put $y = \log_b x$, which is another way of writing $b^{y} = x$. Take the logarithm of both sides in any base $c$: the power rule turns $\log_c(b^{y})$ into $y\log_c b$, so $y\log_c b = \log_c x$, and dividing by $\log_c b$ gives the formula.

The new base $c$ is free to choose. A calculator offers only base $10$ and base $e$, so taking $c$ to be either one evaluates a logarithm in any base with two keys. Better, when $x$ and $b$ are both recognisable powers of some $c$, the top and the bottom are small whole numbers and the answer comes out as an exact fraction.

The formula also says that $\log_b$ and $\log_c$ differ only by the constant factor $\log_c b$, which does not depend on $x$. Changing base never changes the shape of a logarithm; it only rescales it.

Ways to work on it

• Walkthrough. Evaluate an awkward logarithm by rewriting it in a common base.
• Practice. Two steps. The answer is always a non-integer fraction — change of base is the only fast way in.
• Hardest. One step. Spot the right base yourself.

Not sure where to start? Take the ten-question placement test.