The Quotient Rule

Take $\log_{2}\left(\frac{64}{4}\right)$. Since $64 = {2}^{6}$ and $4 = {2}^{2}$, the quotient is ${2}^{4}$: exponents subtract when powers divide. So the log is 4, which is 6 minus 2, the log of the top minus the log of the bottom. That is the quotient rule:

$$\log_{b}\left(\frac{M}{N}\right) = \log_{b}\left(M\right)-\log_{b}\left(N\right)$$

It holds for positive M and N, and the reason is the same as for the product rule. If M is b to the power p and N is b to the power q, then M over N is b to the power p minus q. The order matters: the top's log comes first, the bottom's log is taken away.

With a positive letter on top: $\log_{2}\left(\frac{x}{16}\right)$ becomes $\log_{2}\left(x\right)-\log_{2}\left(16\right)$, and $\log_{2}\left(16\right)$ is 4, so it is $\log_{2}\left(x\right)-4$. Do not divide the logs: the log of a quotient is a difference of logs, not a quotient of logs.