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.

Sorry to use the Answer panel, but I am not allowed to comment. Please, see the answer to this related question for more details on the calculations:
Distributional Laplacian of logarithm and the Dirac delta distribution
