Operator norm, $L^p$ spaces

Question

I have problems with the following task:
Let $(X,\mathcal S,\mu)$ be a $\sigma$-finite measure space. For $g\in L^\infty(X)$ and $p\in[1,\infty)$ look at $M_g:L^p\rightarrow L^p, f\mapsto fg$. I have to calculate $\|M_g\|:=\sup\{\|M_gf\|_p : f \in L^p(X), \|f\|_p \leq 1\}$. Do you have a hint for me how to solve this.