What does $e^{a*ln(x)}$ equal in terms of $a$ and $x$, and how is this found? I saw somewhere that it would be $x^a$, but I'm not sure why.
Only two logarithm rules are needed:
$$\ln(a^b) = b\cdot \ln(a)$$
$$e^{\ln(x)} = x$$
And thus
$$e^{a\cdot \ln(x)} = e^{\ln(x^a)} = x^a$$
$ e^{a*\ln(x)} = e^{\ln(x^a)} = x^a $