Here is the question I am trying to answer:
Let $f:[0,1] \to\Bbb R$ be a Riemann integrable function with $f \ge c>0$. Prove that $$\int_0^1\ln(f(x))\ dx\le \ln\left(\int_0^1 f(x)\ dx\right).$$
I understand how to prove two integrals are equal by showing that their upper and lower Darboux sums are equal and that they converge to the same definite integral. But, I don't understand how to prove the less than or equal to part.
Ideas I've thought about include: integration by parts, improper integrals, partitions.
Does anyone know how to prove this?