Let $\log_{base}(y) = x$ so $\exp_{base}(x) = y$

Here's what that looks like on a mapping diagram:

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:

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 $\log_{base}( y^p ) = p\log_{base}(y)$ |

Martin Flashman, 28 Sept 2014, Created with GeoGebra

So $y = b^p = base^{ \log_{base}(y)}$ and thus