## A Better Change of Base Rule

The change of base rule states
There is a better way to change bases.
We can rewrite
$log_a x^y$
as
$y log_a x$
.
There is a similar rule for changing the base.
$log_{a^b} x = \frac{1}{b} log_a x$

Suppose we want to change the base from an arbitrary base
$u$
to some other arbitrary base
$v$
.
We can use
$u= v^{\frac{ln u}{ln v}}$
where
$ln u$
is the natural logaithm of
$u$
that is
$ln u = log_e u$
where
$e=2.71828...$

Then
$ln_u x = ln_{v^{\frac{ln u}{ln v}}} x = 1/ \frac{ln u}{ln v} ln_v x = \frac{ln v}{ln u} ln_v x$

Use of the base
$e$
here is optional. Any base will do, but natural logs are commonl;y used.
Example: Change the base pf a log vrom 7 to 12.

$log_7 x = log_{12^{ln 7}{ln 12}} x = \frac{ln 12}{ln 7} ln_{12} x$