How to describe the error of the asymptotic approximation of the prime counting function

Question

I've been reading about this asymptotic approximation of the prime counting function $\pi(x)$:$$π(x)=Li(x)+O(\sqrt{x}\ log(x))$$ What does this tell me about the error of this approximation? If the error were just $O(\sqrt{x})$ I would know that if I multiply $x$ by four I would roughly double the error. Is there an as simple interpretation for $O(\sqrt{x}\ log(x))$?

Answer 1

$\log x$ is very small compared to any positive power of $x$, so a good interpretation is "just a bit larger than $O(\sqrt x)$".