Proof

Prove if $x,y,$ and $b$ are positive real values with $b \neq 1$, then

$$\displaystyle{\log_b \frac{x}{y} = \log_b x - \log_b y}$$


First note that we can evaluate $\log_b x$ and $\log_b y$, since $x,y,$ and $b$ are positive real values with $b \neq 1$. Let us denote these two values by $m$ and $n$, so that

$$\log_b x = m \quad \textrm{and} \quad \log_b y = n$$

If we express the above in exponential form, we have

$$b^m = x \quad \textrm{and} \quad b^n = y$$

With the above, the rest of the proof follows quickly, as shown below. Notice how the proof hinges on the fact that $\displaystyle{\frac{b^m}{b^n} = b^{m-n}}$, one of our laws of exponents.

$$\begin{array}{rcl} \log_b \frac{x}{y} &=& \log_b \frac{b^m}{b^n}\\ &=& \log_b b^{m-n}\\ &=& m - n\\ &=& \log_b x - \log_b y \end{array}$$