How to prove that $f(x) = x^ε - \log x$ is $\infty$ when $x\to\infty$?

Question

I'm trying to prove that the function $x^ε$ is "bigger" than $\log x$ when $x\to\infty$, for every $ε>0$.

Or to put it in a more formal way:

For every $ε>0$, there exists a constant $N$ for which all $x>N$ the following is true:

$x^ε > \log x$

I don't believe that the base of the log is important for this one, but just in case, it is 2.

Answer 1

hint: Show $\dfrac{x^{\epsilon}}{\log x} \to +\infty$ by L'hospitale rule.