What does $e^{a*ln(x)}$ equal in terms of $a$ and $x$, and how is this found?

Question

I saw somewhere that it would be $x^a$, but I'm not sure why.

Answer 1

Only two logarithm rules are needed: $$\ln(a^b) = b\cdot \ln(a)$$ $$e^{\ln(x)} = x$$ And thus $$e^{a\cdot \ln(x)} = e^{\ln(x^a)} = x^a$$

Answer 2

$ e^{a*\ln(x)} = e^{\ln(x^a)} = x^a $

Answer 3

$ln(x)$ is the power you need to raise $e$ to, to get $x$ Therefore, $e^{ln(x)}$ = $e$ raised to the power you need to raise e to, to get x. Thus, $e^{ln(x)} = x$

And for $e^{a*ln(x)}$ use the log property: $\ln(a^b) = b\cdot \ln(a)$