Why $1/z * 1/\bar{z} = 1/2\pi \ln r$?

Question

I am learning about fundamental solutions for $P(\partial) = -\Delta$ and the solution is $K_{-\Delta} = \frac{1}{2\pi} \ln r$. One remark from my professor is that $$ K_{-\Delta} = K_\partial * K_{\bar \partial} = \frac{1}{z} * \frac 1 {\bar{z}}. $$ Therefore, we must have $$ \frac{1}{2\pi} \ln r = \frac{1}{z} * \frac 1 {\bar{z}}, $$ where $r = \|z\|$. I am wondering why this is true. Note that $*$ is convolution here.