How to prove that $f(x)$ is much less than $g(x)$?

Question

In order to prove that $f(x) \ll g(x)$, can I prove that $\lim_{x \rightarrow \infty}\dfrac{f(x)}{g(x)}=0$?

Just to clarify my question, I need to prove that $\dfrac{x}{\log(x)-\log(2^t)} \ll \dfrac{x}{4}$, where $t$ is a constant and $x > 2^t$.