How to prove this inequality $$\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$$ for $f>0$.
$\log$ is concave. this is just Jensen's inequality. See http://en.wikipedia.org/wiki/Jensen's_inequality Look at the measure theoretic form. Please check that this makes sense to you.
You can find the proof in https://www.math.ucdavis.edu/~hunter/m218b_09/Lp_and_Sobolev_notes.pdf on page 3.