Question Number 203275 by Frix last updated on 13/Jan/24
$$\mathrm{Show}\:\mathrm{this}\:\mathrm{has}\:\mathrm{exactly}\:\mathrm{7}\:\mathrm{solutions}\:\mathrm{for}\:{x}\in\mathbb{C}: \\ $$$${x}^{\mathrm{ln}\:{x}} =\mathrm{1} \\ $$
Commented by aleks041103 last updated on 14/Jan/24
$${When}\:{you}\:{use}\:{ln}\left({x}\right)\:{do}\:{you}\:{imply}\:{the}\:{principle} \\ $$$${branch}\:{of}\:{this}\:{function}? \\ $$$${Otherwise},\:{ln}\left({x}\right)\:{is}\:{a}\:{multivalued}\:{function}. \\ $$
Commented by Frix last updated on 14/Jan/24
$$\mathrm{We}\:\mathrm{only}\:\mathrm{need}\:\mathrm{the}\:\mathrm{principle}\:\mathrm{branch}\:\mathrm{because} \\ $$$$\mathrm{instead}\:\mathrm{we}\:\mathrm{use}\:\mathrm{1}=\mathrm{e}^{\mathrm{2}\pi{n}\mathrm{i}} \:\mathrm{on}\:\mathrm{the}\:\mathrm{rhs}. \\ $$