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Question Number 203275 by Frix last updated on 13/Jan/24

$$\mathrm{Show}\mathrm{this}\mathrm{has}\mathrm{exactly}7\mathrm{solutions}\mathrm{for}x\in \mathbb{C}:$$$$x^{\ln x}=1$$

Commented by aleks041103 last updated on 14/Jan/24

$$Whenyouuseln\left(x\right)doyouimplytheprinciple$$$$branchofthisfunction?$$$$Otherwise,ln\left(x\right)isamultivaluedfunction.$$

Commented by Frix last updated on 14/Jan/24

$$\mathrm{We}\mathrm{only}\mathrm{need}\mathrm{the}\mathrm{principle}\mathrm{branch}\mathrm{because}$$$$\mathrm{instead}\mathrm{we}\mathrm{use}1={\mathrm{e}}^{2\pi n\mathrm{i}}\mathrm{on}\mathrm{the}\mathrm{rhs}.$$

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