TITLE: Does the integrability of $\log(f(x))$ imply $f(x)$ is bounded?
QUESTION [6 upvotes]: Let $f(x):(a,b)\rightarrow \mathbb{R}$. Suppose $\int_a^b\log{f(x)}\,\mathrm{d}x<\infty$, can we claim that $0<f(x)<M<\infty$ a.e.. Why and why not?

REPLY [5 votes]: Hint: Consider $f\colon (0,1)\to \Bbb R, x\mapsto \dfrac{1}{x}$.