Question Number 5591 by FilupSmith last updated on 21/May/16
$$\mathrm{For}\:{x}^{{x}^{{x}^{...} } } ={a} \\ $$$$\therefore\:{x}^{{a}} ={a}\:\:\:\Rightarrow\:\:\:{x}={a}^{\frac{\mathrm{1}}{{a}}} \\ $$$$ \\ $$$$\mathrm{Is}\:\mathrm{this}\:\mathrm{true}? \\ $$$$\mathrm{When}\:\mathrm{is}\:\mathrm{it}\:\mathrm{not}\:\mathrm{true}? \\ $$
Commented by prakash jain last updated on 21/May/16
$$\mathrm{If}\:\mathrm{the}\:\mathrm{LHS}\:\mathrm{exists}\:\mathrm{and}\:\mathrm{is}\:\mathrm{finite}\:\mathrm{the}\: \\ $$$${x}^{{a}} ={a}. \\ $$