$$e^{1/z}=-1$$

From the get go, I can see that if $z$ is a purely real number, this statement can't be true. But I need to show what happens with complex numbers, so below is my attempt. 

My question is: is it ok to leave the final answer in the notation below? I know we normally don't leave $i$ in the denominator so I removed it, but then this answer seems kind of sloppy...but then again, I'm not sure.

Taking the natural log of both sides yields:

$$\frac{1}{z}=i\pi + 2\pi i k$$

$$z=\frac{1}{i\pi + 2\pi i k}$$

Rationalizing the denominator:

$$z=\frac{2ki-i}{\pi -4\pi k^2}$$