Proof ELF.APP.3 : $\log_{base}( y^p ) = p\log_{base}(y) $
  Proof : We begin by using the inverse relation of  $\log_{base}$ and $\exp_{base}$ for $y$.
Let $\log_{base}(y) = x$ so  $\exp_{base}(x) = y$

Here's what that looks like on a mapping diagram:MD exp log notation
Next we consider the power:
 $y^p = (base^x)^p = base^{px}$
and thus we have the corresponding logarithmic equation:
$\log_{base} (y^p) = \log_{base} (base^{px}) = px = p\log_{base}(y)$

Here's what that looks like on a mapping diagram:Md for Prod with log

And here's a mapping diagram showing the product of the logarithm visualized as well:
MD for the entire proof
Finally here is a dynamic visualization of the proof using mapping diagrams:
Proof of $\log_{base}( y^p ) = p\log_{base}(y)$

Martin Flashman, 28 Sept 2014, Created with GeoGebra

Corollary: $ \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
$\ln(y) = \log_{base}(y) \cdot \ln(base).$