How to solve ${\rm Ln}({\rm cosecant}(z)) = {\rm Ln}(2) + iz$?

Question

$\newcommand{\Ln}{\operatorname{Ln}}$ $$\Ln(\csc(z)) = \Ln(2) + iz$$ I need to know what complex number $z$ is

Answer 1

Hint

$$\dfrac{\csc(z)}2=e^{iz}$$

If $e^{iz}=a,$

Using Intuition behind euler's formula

$2a=\dfrac{2i}{a-1/a}$ as $a\ne0$

$a^2-1=i$

$a^2=\sqrt2e^{i\pi/4}$