Continuity of log(x) as a multiplication operator

Question

Define a multiplication operator $T_{log}f(x) = f(x)\cdot log(x)$. For $1 \le p < \infty$ and the usual Lebesgue space $L^p(\mathbb{R})$, is $T_{log}$ a continuous operator from $L^p(\mathbb{R})$ to itself? I know this is not true for $p = \infty$ and I feel this should be true for finite $p$. But I don't know how to prove it or if there exist counter examples.