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

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

Next we consider the product:

$y\cdot b = base^x \cdot base^a = base^{x+a}$

and thus we have the corresponding logarithmic equation:

$\log_{base} (y\cdot b) = \log_{base} (base^{x+a}) = x+a = \log_{base}(y) + \log_{base}(b)$

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

And here's a mapping diagram showing the sum of the logarithms visualized as well:

Finally here is a dynamic visualization of the proof using mapping diagrams:

Proof of $\log_{base}( y\cdot b ) = \log_{base}(y) + \log_{base}(b)$ |

Martin Flashman, 20 Sept 2014, Created with GeoGebra