Let $g$ be a non-negative measurable function on $[0,1]$. How can I show that $ \int \log ~(g(u))~\text{d}u \leq \log~\int g(u)~\text{d}u $ whenever the left hand side is defined.
If it helps, I know that the $\log$ function is concave.
Edit:
Can I argue like this: Since $\log$ concave, $-\log$ is convex; So by Jensen's inequality, we have $ -\log~\int g(u)~\text{d}u ~\leq~ \int-\log ~(g(u))~\text{d}u?$