(1)

\begin{align*} \Log e^{z} & =\Log e^{\Re z+i\Im z}\\ & =\ln e^{\Re z}+\Log e^{i\Im z}\\ & =\Re z+\ln\left|e^{i\Im z}\right|+i\Arg e^{i\Im z}\\ & =\Re z+i\mod\left(\Im z,-2\pi,\pi\right) \end{align*}

(2)

\begin{align*} \Log\left|\alpha\right|^{\beta} & =\Log\left|\alpha\right|^{\Re\left(\beta\right)+i\Im\left(\beta\right)}\\ & =\ln\left|\left|\alpha\right|^{\Re\left(\beta\right)+i\Im\left(\beta\right)}\right|+i\Arg\left(\left|\alpha\right|^{\Re\left(\beta\right)+i\Im\left(\beta\right)}\right)\\ & =\ln\left|\left|\alpha\right|^{\Re\left(\beta\right)}\left|\alpha\right|^{i\Im\left(\beta\right)}\right|+i\Arg\left(\left|\alpha\right|^{\Re\left(\beta\right)}\left|\alpha\right|^{i\Im\left(\beta\right)}\right)\\ & =\ln\left|\alpha\right|^{\Re\left(\beta\right)}+i\Arg\left(\left|\alpha\right|^{i\Im\left(\beta\right)}\right)\\ & =\ln\left|\alpha\right|^{\Re\left(\beta\right)}+i\Arg\left(e^{i\Im\left(\beta\right)\ln\left|\alpha\right|}\right)\\ & =\Re\left(\beta\right)\ln\left|\alpha\right|+i\mod\left(\Im\left(\beta\right)\ln\left|\alpha\right|,-2\pi,\pi\right) \end{align*}

(2)-2

\begin{align*} \Log\left|\alpha\right|^{\beta} & =\Log\left|\alpha\right|^{\left|\beta\right|e^{i\Arg\left(\beta\right)}}\\ & =\ln\left|\left|\alpha\right|^{\left|\beta\right|e^{i\Arg\left(\beta\right)}}\right|+i\Arg\left(\left|\alpha\right|^{\left|\beta\right|e^{i\Arg\left(\beta\right)}}\right)\\ & =\ln\sqrt{\left|\alpha\right|^{\left|\beta\right|e^{i\Arg\left(\beta\right)}}\left|\alpha\right|^{\left|\beta\right|e^{-i\Arg\left(\beta\right)}}}+i\Arg\left(\left|\alpha\right|^{\left|\beta\right|e^{i\Arg\left(\beta\right)}}\right)\\ & =\ln\sqrt{\left|\alpha\right|^{\left|\beta\right|\left(e^{i\Arg\left(\beta\right)}+e^{-i\Arg\left(\beta\right)}\right)}}+i\Arg\left(e^{\left|\beta\right|e^{i\Arg\left(\beta\right)}\Log\left|\alpha\right|}\right)\\ & =\ln\sqrt{\left|\alpha\right|^{2\left|\beta\right|\cos\left(\Arg\beta\right)}}+i\Arg\left(e^{\left|\beta\right|\Log\left|\alpha\right|\left(\cos\left(\Arg\beta\right)+i\sin\left(\Arg\beta\right)\right)}\right)\\ & =\ln\left|\alpha\right|^{\left|\beta\right|\cos\left(\Arg\beta\right)}+i\Arg\left(e^{\left|\beta\right|\Log\left|\alpha\right|\cos\left(\Arg\beta\right)+i\left|\beta\right|\Log\left|\alpha\right|\sin\left(\Arg\beta\right)}\right)\\ & =\left|\beta\right|\cos\left(\Arg\beta\right)\ln\left|\alpha\right|+i\Arg\left(e^{\left|\beta\right|\Log\left|\alpha\right|\cos\left(\Arg\beta\right)}e^{i\left|\beta\right|\Log\left|\alpha\right|\sin\left(\Arg\beta\right)}\right)\\ & =\left|\beta\right|\cos\left(\Arg\beta\right)\ln\left|\alpha\right|+i\mod\left(\left|\beta\right|\sin\left(\Arg\beta\right)\Log\left|\alpha\right|,-2\pi,\pi\right)\\ & =\Re\left(\beta\right)\ln\left|\alpha\right|+i\mod\left(\Im\left(\beta\right)\Log\left|\alpha\right|,-2\pi,\pi\right) \end{align*}

(3)

\begin{align*} \Log\alpha^{\beta} & =\Log e^{\beta\Log\alpha}\\ & =\Log e^{\left(\Re\left(\beta\right)+i\Im\left(\beta\right)\right)\left(\ln\left|\alpha\right|+i\Arg\left(\alpha\right)\right)}\\ & =\Log\left(e^{\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\Arg\left(\alpha\right)}e^{i\left(\Re\left(\beta\right)\Arg\left(\alpha\right)+\Im\left(\beta\right)\ln\left|\alpha\right|\right)}\right)\\ & =\ln e^{\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\Arg\left(\alpha\right)}+i\Arg e^{i\left(\Re\left(\beta\right)\Arg\left(\alpha\right)+\Im\left(\beta\right)\ln\left|\alpha\right|\right)}\\ & =\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\Arg\left(\alpha\right)+\mod\left(\Re\left(\beta\right)\Arg\left(\alpha\right)+\Im\left(\beta\right)\ln\left|\alpha\right|,-2\pi,\pi\right) \end{align*}