Prove $5z^n = e^z$ has no solutions on the annulus $1 < \lvert z \rvert < 2$ for any $n \ge 1$.
If $z = a+bi$ then $5z^n = 5\lvert z \rvert^ne^{in\theta}$ and $e^z = e^ae^{bi}$.
Direction to a useful theorem(s) or techniques for this would be nice. I am in the process of teaching myself complex analysis.