Using concavity of log to prove $L^p$ norm is increasing in $p$

Question

In this proof that $p \mapsto \|f\|_p$ is an increasing function of $p > 0$ (for fixed $f \in L^p$ on a finite measure space) the approach is to show that $\frac{d}{dp} \|f\|_p \geq 0$, which ends up being equivalent to $$ \int |f|^p \ln|f|^p \geq \ln \int |f|^p. $$ It's stated in the final paragraph that one can approximate the integrals by finite sums and use the concavity of $\ln$ to prove that inequality. I'm having trouble with this though. Can someone sketch this out in more detail to help me understand?

Answer 1

First, lets assume that we have a finite measure space with measure $\mu(\Omega)=1$. Because if $\mu(\Omega)>1$ then the claim is wrong, using $f=1$ as a counterexample.

Then the last equation from the linked pdf $$ \int |f|^p \ln |f|^p \geq \int |f|^p \ln \int |f|^p $$ is true because the function $g(x)= x \ln(x)$ is convex and then we can directly apply Jensen's Inequality.