## A Better Change of Base Rule

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.

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