Solve $\displaystyle |e^z| = 2$ for z

Attempt:

$\displaystyle \sqrt{e^{2z}} = e^z$

Therefore:

$\displaystyle e^z = 2e^{0 + i2\pi k}$

$\displaystyle = e^{ln2 + i2\pi k}$

$\displaystyle z = ln2 +i2\pi k, k E Z$

but the correct answer was $\displaystyle z = ln2 + ik$