Change of Base Theorem

This is COB - Change Of Base Any logarithm can be expressed in terms of common logarithm

Theorem

$\frac{\log t}{\log b}={\log }_{b}t=x$

Proof

$x={\log }_{b}m$

1. Rewrite as an exponential $b^x=m$
2. log both sides $\log b^x=\log m$
3. Power law. bring the moon down to earth $x\log b=\log m$
4. solve for x x = $\frac{{\log }_{anybase}m}{{\log }_{anybase}b}$

Example

${\log }_2(9)=x$ $x=\frac{{\log }_9}{{\log }_2}$ you can leave it in that form, or turn it into decimal $x=3.16992500144$