A Better Change of Base Rule

The change of base rule states
There is a better way to change bases.
We can rewrite  
\[log_a x^y\]
\[y log_a x\]
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  
  to some other arbitrary base  
We can use  
\[u= v^{\frac{ln u}{ln v}}\]
\[ln u\]
  is the natural logaithm of  
  that is  
\[ln u = log_e u\]

\[ln_u x = ln_{v^{\frac{ln u}{ln v}}} x = 1/ \frac{ln u}{ln v} ln_v x = \frac{ln v}{ln u} ln_v x\]

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

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