Proof ELF.APP.2

ELF.APP.2 : $\log_{base}( \frac yb ) = \log_{base}(y) - \log_{base}(b)$
Proof : We begin by using the inverse relation of $\log_{base}$ and $\exp_{base}$ for $y$ and $b$.
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: MD exp log notation

Next we consider the quotient:

$\frac yb = base^x / base^a = base^{x-a}$

and thus we have the corresponding logarithmic equation:

$\log_{base} (\frac yb) = \log_{base} (base^{x-a}) = x-a = \log_{base}(y) - \log_{base}(b)$

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

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

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

Proof of $\log_{base}( \frac yb) = \log_{base}(y) - \log_{base}(b)$
Exponential And Logarithmic Functions - GeoGebra Dynamic Worksheet

Martin Flashman, 21 Sept 2014, Created with GeoGebra