Power rule for Logarithms
Solution 1:
If you don't mind using calculus, start with $$ \ln(x) = \int_1^x \frac{dt} t .$$ Make the substitution $t = s^a$, so $dt = a s^{a-1} ds$, and you get $$ \ln(x^a) = \int_1^{x^a} \frac{dt} t = a \int_1^{x} \frac{ds} s = a \ln(x).$$