If $\mathrm{\omega }$ is an Imaginary fifth root of unity then log ${\log }_2\left|1+\mathrm{\omega }+{\mathrm{\omega }}^2+{\mathrm{\omega }}^3-\frac{1}{\mathrm{\omega }}\right|$

$\begin{array}{rcl}{\mathrm{w}}^5& =& 1, 1+\mathrm{w}+{\mathrm{w}}^2+{\mathrm{w}}^3+{\mathrm{w}}^4=0\end{array}$

$\begin{array}{rcl} & & \log \left|1+\mathrm{w}+{\mathrm{w}}^2+{\mathrm{w}}^3-\frac{1}{\mathrm{w}}\right|\end{array}$

$\begin{array}{rcl} & & \log \left|1+\mathrm{w}+{\mathrm{w}}^2+{\mathrm{w}}^3-{\mathrm{w}}^4\right|\end{array}$

$\begin{array}{rcl} & & \log \left|\left(1+\mathrm{w}+{\mathrm{w}}^2+{\mathrm{w}}^3+{\mathrm{w}}^4\right)-2{\mathrm{w}}^4\right|\end{array}$

$\begin{array}{rcl}{\log }_2 \left|\left(2{\mathrm{w}}^4\right)\right|& =& {\log }_2 \left|2\right|=1\end{array}$