Change of Base of Logarithm

Theorem
Let $\log_a x$ be the logarithm to base $a$ of $x$.

Then:
 * $\log_b x = \dfrac {\log_a x} {\log_a b}$.

Thus a convenient formula for calculating the logarithm of a number to a different base.

Proof
Let:
 * $y = \log_b x \iff b^y = x$
 * $z = \log_a x \iff a^z = x$

Then:

Hence the result.

Note
Some people prefer to write this as:
 * $\log_a x = \log_a b \ \log_b x$

as it is delightfully easy to remember.