real analysis - Proving that $f(x)=frac{1}{x^2 ln x} $ is Lebesgue measurable on $(2, + infty)$

I have that a set $E$ is Lebesgue measurable if the outer measure:
$$\mu^*(E)=\inf_{I_1,...,I_n} \mu (I), E \subseteq I_1 \cup I_2 ,...\cup I_n , I_i-\text{intervals}$$

satisfy the three properties of measure.
Proving that $$f(x)=\frac{1}{x^2 \ln x} $$ is Lebesgue measurable would suggest that I must prove that the set $f((2,+\infty))$ satisfies these properties. Is this correct, and how do I go about proving this? Maybe using the fact that the function is a continuous bijection and a map of an open set would also be open and this would be a subset of Borel- sigma algebra. Help ?