The Power Rule

Take $\log_{2}\left(8^{2}\right)$. Since $8 = {2}^{3}$, the power is ${\left({2}^{3}\right)}^{2}$, which is ${2}^{6}$: a power of a power multiplies the exponents. So the log is 6, which is 2 times 3, the exponent times the log of 8. That is the power rule:

$$\log_{b}\left(M^{k}\right) = k\log_{b}\left(M\right)$$

It holds for positive M, and the exponent comes down in front of the log. The reason: if M is b to the power p, then M to the power k is b to the power kp. You can also see it as the product rule repeated: $\log_{b}\left(M^{3}\right)$ is the log of M times M times M, which is three logs of M added.

With a letter, watch the sign. For x > 0, $\log_{3}\left(x^{4}\right)$ becomes $4\log_{3}\left(x\right)$. At x = -1 the left side is $\log_{3}\left(1\right)$, which is 0, but $\log_{3}\left(-1\right)$ has no value, so the rule needs x positive. The rule works in every base, base 10 included.