$ \log ||f||_p=\log \left(\int |f|^p\right)^{1/p}=\frac{1}{p}\log\left(\int|f|^p\right)\geq\frac{1}{p}\int\log|f|^p=\int\log|f|d\mu $
The last but two inequality is due to the integral form of Jensen's inequality.
$ \log||f||_p=\frac{1}{p}\log\left(\int|f|^p\right)\leq\frac{1}{p}\left(\int|f|^p-1\right)=\frac{1}{p}\int(|f|^p-1)=\int\frac{|f|^p-1}{p} $
First inequality is by $ log(x)\leq x-1 $ from hint; the second equality is due to the property of probability space$ \int d\mu=1 $
By L' Hospital Rule,$ \phi(p)=\frac{|f|^p-1}{p}\to log|f|,(p\to0) $
Since $ \phi(p) $ is monotone increasing, by Monotone Convergence Theorem, $ \int\frac{|f|^p-1}{p}\to\int\log|f|d\mu $
Therefore,$ \lim\limits_{p\to0}\log||f||_p=\int_X\log|f|d\mu $, and the result follows by using Exp on the both sides.
