Change of Base of Logarithm

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

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

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.