Answer
See next section for a step-by-step proof of the quotient rule.
Work Step by Step
We wish to prove that $$\log_{b}(\frac{M}{N}) = \log_{b}M - \log_{b}N$$ To that end, let $\log_{b}M = R$ and $\log_{b}N = S$: $$\log_{b}(M) = R → b^R = M$$ $$\log_{b}N = S → b^S = N$$ Therefore, $$\frac{M}{N} = \frac{b^R}{b^S}$$ $$\frac{M}{N} = b^{R-S}$$ Changing this last equation to logarithmic form, we arrive at the following: $$b^{R-S} = \frac{M}{N}$$ $$log_{b}(\frac{M}{N}) = R – S$$ And, by substituting for the original values of $R$ and $S$: $$log_{b}(\frac{M}{N}) = \log_{b}M - \log_{b}N$$
