For what complex numbers w does the equation exp(z)=w have solutions? For what complex numbers w does the equation $\exp(z)=w$ have solutions?
I'm working through George Cain's free Complex Analysis textbook.  Love that it's free, but there are no solutions to the exercises.
Solving the equation, I found that $y=\tan^{-1}(\frac{v}{u})$ and $x=\ln\lvert w 
\rvert$
Looks like the answer is $u\neq0$.  That will satisfy the issues in both equations.
Problem is, I was able to solve $\exp(z)=ai$.  the solution is $x=\ln a+i(\frac{\pi}{2}+2k\pi)$.  This kind of makes sense because $\lim_{x\to\infty}\tan^{-1}x=\frac{\pi}{2}$.
The algebra is breaking down when I solve $e^x\cos y=u$ for $x$.  You can't divide by $\cos y$ if it ends up being zero.  
Long story short...I'm pretty sure the answer is $w\neq0$, but I'm looking for a more elegant approach to getting there.  Any help?

If $z=x+yi$ then $\exp(z)=e^x e^{yi}$. In particular $|\exp(z)|=e^x$.
If $w\ne0$, then take $x=\ln |w|$ and $y$ such that $e^{yi}=\frac{w}{|w|}$.
More precisely, write $w=r e^{i\theta}$. Then $x=\ln r$ and $y=\theta$ work for $\exp(z)=w$.
