ELF.AP.Proof.4 : $ \ln(y) = \log_{base}( y ) \cdot ln(base) $ or $\log_{base}(y) = \frac{\ln(y)} {\ln(base)}$ Proof : Apply $\ln(b^p)=p\ln(b)$ with $b = base ; p = \log_{base}(y)$.
So $y = b^p = base^{ \log_{base}(y)}$ and thus $\ln(y) = \log_{base}(y) \cdot \ln(base).$
This can also be proven directly by using the inverse relation of $\ln $ and $ \exp_{e}$ for the $base$.
Let $\ln(base)=a$ so $e^a=base$.
Here's what that looks like on a mapping diagram:
Next we consider $x= \log_{base} (y)$ so $base^x = y$.
Then $(e^a)^x =e^{ax} = y $ and thus we have the corresponding logarithmic equation:
$\ln(y)=ax= \ln(base) \cdot \log_{base}(y)$.
Here's what that looks like on a mapping diagram:
And here's a mapping diagram showing the product of the logarithm visualized as well:
Finally here is a dynamic visualization of the proof using mapping diagrams:
Proof of $ \ ln(y) = \log_{base}( y ) \cdot ln(base) $
Martin Flashman, 26 Oct 2014, Created with GeoGebra