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