How do I prove that function is solution of the Laplace equation?

Question

How do I prove that for $\vec{r}=(x,y,z)\in \mathbb{R}^3,\vec{r}\neq 0$, function is $u(x,y,z):=1/(-ln\left \| \vec{r} \right \|)$ a solution of the Laplace equation $\Delta u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}=0$?

Answer 1

Just like for any differential equation, you check whether it's a solution by plugging the alleged solution into the equation and see whether the difference of the two sides simplifies to $0$. Here it doesn't, so this is not a solution.