Find all possible values of $z$ such that $\cos z=\frac{3}{4}+\frac{i}{4}$.
Here is my attempt:
$\cos z=\frac{\mathit{e}^{zi}+\mathit{e}^{-zi}}{2}=\frac{3}{4}+\frac{i}{4}$
$2\mathit{e}^{zi}+2\mathit{e}^{-zi}=3+i$
$2\mathit{e}^{2zi}+2=3\mathit{e}^{zi}+i\mathit{e}^{zi}$
Rearranging and using the quadratic formula, I get:
$\mathit{e}^{zi}=\frac{3+i\pm \sqrt{-6-6i}}{4}$
The answer in the back of the textbook is $z=\pm (\frac{\pi}{4}+2\pi n-i\frac{1}{2}\log 2)$. But after logging both sides of my equation and dividing each side by $i$, I do not get the same $z$. Is this because my solution is wrong or is it because I simplified wrong?
Any tips are appreciated!