The Product Rule

Take $\log_{3}\left(9\cdot 27\right)$. Since $9 = {3}^{2}$ and $27 = {3}^{3}$, the product is ${3}^{2}\cdot {3}^{3}$, which is ${3}^{5}$: exponents add when powers multiply. So the log is 5, which is 2 plus 3, the log of 9 plus the log of 27. That is the product rule:

$$\log_{b}\left(MN\right) = \log_{b}\left(M\right)+\log_{b}\left(N\right)$$

It holds for any positive M and N, for the same reason. Logs are exponents: if M is b to the power p and N is b to the power q, then MN is b to the power $p+q$. So the log of the product is the sum of the logs.

The rule is most useful when one factor is a letter. For positive x, $\log_{3}\left(9x\right)$ becomes $\log_{3}\left(9\right)+\log_{3}\left(x\right)$, and $\log_{3}\left(9\right)$ is 2, so it is $2+\log_{3}\left(x\right)$.