The value of z satisfying the equation $\log \mathrm{z}+\log {\mathrm{z}}^2+\cdots +\log {\mathrm{z}}^{\mathrm{n}}=0$ is

We have

$\begin{array}{l}\log \mathrm{z}+\log {\mathrm{z}}^2+\log {\mathrm{z}}^3+\cdots +\log {\mathrm{z}}^{\mathrm{n}}=0\\ \Rightarrow \log \left({\mathrm{zz}}^2{\mathrm{z}}^3\cdots {\mathrm{z}}^{\mathrm{n}}\right)=0\\ \Rightarrow \log \left({\mathrm{z}}^{\frac{\mathrm{n}(\mathrm{n}+1)}{2}}\right)=0\end{array}$

$\begin{array}{c}\mathrm{or} {\mathrm{z}}^{\frac{\mathrm{n}(\mathrm{n}+1)}{2}}=1\\ \mathrm{or} \mathrm{z}=1^{\frac{2}{\mathrm{n}(\mathrm{n}+1)}}\\ ={\left(\cos 0^{\circ }+\mathrm{isin}{0}^{\circ }\right)}^{\frac{2}{\mathrm{n}(\mathrm{n}+1)}}\\ =(\cos 2\mathrm{m\pi }+\mathrm{isin}2\mathrm{m\pi }{)}^{\frac{2}{\mathrm{n}(\mathrm{n}+1)}},\mathrm{m}=0,1,2,3,\dots \\ =\cos \frac{4\mathrm{m\pi }}{\mathrm{n}(\mathrm{n}+1)}+\mathrm{isin}\frac{4\mathrm{m\pi }}{\mathrm{n}(\mathrm{n}+1)},\mathrm{m}=0,1,2,\dots \end{array}$