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Can someone please tell me if I am approaching this correctly? Given the following and asked to solve for the complex variable z: $$[e^z]^3-5e^z=0$$

My approach was purely algebraic and is why I have my doubts: $$[e^z]^3=5e^z$$ $$[e^z]^2=5$$ $$z=\frac{\ln 5}{2}+i0$$

Did I over simplify or overlook something? Or is it this simple?

NotSoSN
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1 Answers1

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Until $e^{2z} = 5$ ok, but the complex exponential is periodic with period $2\pi i$, so we actually get: $$2z =\ln 5 + 2k\pi i, \quad k \in \Bbb Z \implies z = \frac{\ln 5}{2} + k\pi i,\quad k \in \Bbb Z.$$

Ivo Terek
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  • Got it, so just like when using the exponential or polar form, I need to make sure I account all angles? – NotSoSN Feb 12 '16 at 03:56
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    Sort of. The issue is that the incorrect step $e^{2z} = 5 \implies 2z = \ln 5$ actually assumes that we have a well defined inverse $\ln$ to apply on both sides, but since the exponential is periodic, we don't have the complex logarithm defined everywhere. Then we start worrying about branches, etc. Each choice of $k$ corresponds to one choice of logarithm. – Ivo Terek Feb 12 '16 at 04:00