Let $u(w)$ be a compact support, real valued, smooth function on $\mathbb C$(hence also on $\mathbb R^2$), can we define $\eta(z) = \int_{\mathbb {R^2}} \log(|z-w|) \, u(w) \, \mathrm{d}x \mathrm{d}y $ for all $z\in \mathbb C$?
When $w=z$, $\log(0)=- \infty$, so the inside of $\eta(z)$ has a singularity, but it seems to me $\eta$ is well defined when reading my notes.
Since $u$ has compact support and is bounded, it suffices to show that $$ \int_{B_1(w)} \lvert \log |z-w| \rvert \, dxdy $$ is finite, where $B_1(w)$ is the ball with center $w$ and radius $1$. With polar coordinates $z = w + re^{i \varphi}$ this becomes $$ \int_{0}^{2\pi} \int_0^1 \lvert \log r \rvert r \, dr \, d\varphi = - 2 \pi \int_0^1 r \, \log r \, dr $$ and that is finite because $r \, \log r$ has a finite limit at $r=0$, i.e. it can be extended to a continuous function on $[0, 1]$.