How to tell if this function is not in$ L^∞ (E)$space?

Question

I let $E＝(0,1)$ and let $f(x)＝ln(\frac{1}{x})$. Is that $f(x)∈L^∞(E)$ I tried to use $|lnx|$ equals $|ln(\frac{1}{x})|$ and then use $|x-1|$ to shrink it, but I couldn't get the answer I was looking for and who could give me some help

Answer 1

$f\notin L^∞(E)$:

Suppose $|\ln (\frac 1 x)| \leq M$ a.e. Then $|\ln (\frac 1 x)| \leq M$ on a dense set of points and, by continuity, the inequality holds for all $x \in (0,1)$. You ge a contradiction by taking $0<x< e^{-M}$.