example of $f\in L^2(B)$ but $f(x)|x|^{-1}\notin L^1(B)$

Question

Let $B$ denote the unit disk in $\mathbb{R}^2$. Find a function $f\in L^2(B)$ such that $f(x)|x|^{-1}\notin L^1(B)$.

I tried $|x|^{-1}\log|x|$ and $(|x|\log|x|)^{-1}$, but they did not work.

Answer 1

Hint: Try $$ f(x)=(\log|x/2|)^{-1}|x|^{-1} $$ That should fix up any problems you might have had.