How to prove this relation between the laplacian of the logarithm and the dirac delta function? Why is this true in two dimensions?
$$\nabla^2\bigg(\ln(r)\bigg)=2\pi\delta^{(2)}(\mathbf{r}),$$
where $\delta^{(2)}$ denotes the two-dimensional $\delta$-function and $r=\sqrt{x^2+y^2}$ in Cartesian coordinates.
I understand that both function will look the same. But I do not know how to prove this rigorously.

Hint: Let  $\phi \in C^\infty_0(\mathbb{R}^2)$. You need to show
\begin{align}
\int_{\mathbb{R}^2} \log|x| \Delta \varphi(x)\ dx = 2\pi \varphi(0).
\end{align}
This can be done by splitting the left-hand side into 
\begin{align}
\int_{B(0, \epsilon)} \log|x| \Delta \varphi(x)\ dx + \int_{\mathbb{R}^2\backslash B(0, \epsilon)} \log |x| \Delta \varphi(x)\ dx = I_1+I_2.
\end{align}
For $I_2$, use integration by parts to put the Laplacian on to $\log |x|$ (don't forget the boundary term).  For $I_1$, show it vanishes as $\epsilon\rightarrow 0$.  Remember the boundary term? Show that converges to the desired quantity as $\epsilon \rightarrow 0$.
