Given is $f: \Re^2 \backslash \{0\} \to \Re $ such that for all $x \in \Re^2 \backslash \{0\}, f(x) = \log|x|$ Show that $f$ is harmonic. This is the solution but I can't understand it. \begin{align} {\partial^2}_{x_i} \log|x|= \frac{1}{2}{\partial^2}_{x_i} \log|x|^2 = \partial_{x_i} \left( \frac{x_i}{|x|^2} \right)= \frac{1}{|x|^2}-\frac{2x_i^2}{|x|^4} \end{align} so it follows \begin{align} \Delta \log|x| = \frac{2}{|x|^2} - 2 \sum_{i=1}^{2} \frac{x_i^2}{|x|^4}=\frac{2}{|x|^2}-\frac{2}{|x|^2}=0 \end{align}
The whole point is to show that $\Delta f$ is $0$. But I don't get even the second step. $\frac{1}{2}{\partial^2}_{x_i} \log|x|^2 = \partial_{x_i} \left( \frac{x_i}{|x|^2} \right)$
It is $\log(|x|^2)$, not $(\log|x|)^2$. $$ \frac{\partial}{\partial x_i}\log(|x|^2)=\frac{\partial}{\partial x_i}\log(x_1^2+x_2^2)=\frac{2\,x_i}{|x|^2}. $$