Finding arcsin(2) using complex exponentials

[Craptola]

Got a maths exam tomorrow been looking through some past papers. Have hit a stumbling block with regard to complex numbers, the problem lies with my algebra.

Homework Statement

Show that $\arcsin(2) = \frac{\pi}{2}-i\ln (2\pm \sqrt3)$

Homework Equations

I'm fairly certain the way to solve this is to use
$$\sin(z)=\frac{1}{2i}(e^{iz}- e^{-iz})$$

The Attempt at a Solution

Equating sin(z) to 2 I could only rearrange it to

$$4i=e^{iz}-e^{-iz}$$
I was always pretty awful at algebra and can't see a way to rearrange for z. If anyone could nudge me in the right direction it would be greatly appreciated.

 

[Hypersphere]

The problem becomes simpler if you make the substitution $v=e^{iz}$.

 

[haruspex]

When the question says "show that a=b", it gives you the option of working from the other end.