Integrability of logarithm

Question

Let $u\in L^p(\Omega)$ for $p>1$ and $\Omega$ be a bounded domain in $\mathbb{R}^N$. Then $\ln u\in L^p(\Omega)$.

Can somebody help me proving this fact. I have used the fact that $e^{\ln u}=u$ which gives $\ln u\in L^p$ if $\ln u>0$. But when $\ln u<0$, I have no hint.

Thanks.

Answer 1

$u \equiv 0$ is in $L^{p}$ but $\ln u\equiv -\infty$ is not in $L^{p}$.

An example with $u>0$: let $\Omega =(0,1)$ and $u(x)=e^{-\frac 1 {\sqrt x}}$. The $u \in L^{2}$ but $\ln u \notin L^{2}$