Values of the complex logarithm and complex powers

If $a$ and $b$ are complex numbers and $a \neq 0$, the set $a^{b}$ consists of those complex numbers $c$ having a logarithm of the form $b\alpha$, for some logarithm $\alpha$ of $a$. (That is, $e^{b\alpha } = c$ and $e^{\alpha } = a$ for some complex number $\alpha$.) Describe set $a^{b}$ when $a = 1$ and $b = 1/3+i$.

Write all values of $i^{i}$ in the form $a+bi$.

Find the value of $\ln (-e)$.

Find the value of $\ln (-2)$.

Find the value of $\ln i$.

Find the value of $\ln \dfrac {1+i}{\sqrt{2}}$.

Find the value of $\ln (x+iy)$.

Find the value of $e^{\pi i}$.

Find the value of $i^{i}$.

Find the value of $2^{i}$.

Find the value of $\left(\dfrac {1+i}{\sqrt{2}}\right)^{-i}$.

Find the value of $\arctg xi$.