Question Number 113954 by Khalmohmmad last updated on 16/Sep/20
$${x}^{{y}} ={y}^{{x}} \:\:\:\: \\ $$$$\begin{cases}{\mathrm{x}}\\{\mathrm{y}}\end{cases}=? \\ $$
Commented by Dwaipayan Shikari last updated on 16/Sep/20
$${when}\:{x}={y} \\ $$$${x},{y}\in\mathbb{Z} \\ $$
Answered by $@y@m last updated on 16/Sep/20
$$\mathrm{2},\mathrm{4} \\ $$
Answered by Dwaipayan Shikari last updated on 16/Sep/20
$$\frac{{x}}{{logx}}=\frac{{y}}{{logy}}={k}\:\:\left({k}\neq\mathrm{0}\right)\:\:\:\left({x}^{{y}} ={y}^{{x}} \Rightarrow{xlogy}={ylogx}\right) \\ $$$${x}={klogx} \\ $$$${e}^{{logx}} ={klogx} \\ $$$${e}^{−{logx}} =\frac{\mathrm{1}}{{klogx}} \\ $$$$−{logxe}^{−{logx}} =−\frac{\mathrm{1}}{{k}} \\ $$$$−{logx}={W}_{\mathrm{0}} \left(−\frac{\mathrm{1}}{{k}}\right) \\ $$$${x}={e}^{−{W}_{\mathrm{0}} \left(−\frac{\mathrm{1}}{{k}}\right)} \\ $$$${y}={e}^{−{W}_{\mathrm{0}} \left(−\frac{\mathrm{1}}{{k}}\right)} \\ $$$${k}\in\left({any}\:{number}\right)\left({k}\neq\mathrm{0}\right) \\ $$