I've seen equations like $\sin^2(x)$ or $\ln^2(x)$, but what does the power do to this? I've tried looking the meaning of this up, but everything I get involves properties of powers that I already know about, such as $2\log(x)\equiv\log(x)^2$.
So what is the difference between say, $\log^2(x)$ and $\log(x)^2$?
It just a notational convention.
By $\sin^2(x)$, we mean $(\sin(x))^2$, but the latter is more cumbersome to write.
The problem with writing $\sin(x)^2$ is that the reader might think it means $\sin(x^2)$.
Similarly, the expression $\log^2(x)$ means $(\log(x))^2$, not $\log(x^2)$.