Question Number 192171 by mathlove last updated on 10/May/23
$$\mathrm{2}^{{x}^{{x}^{{x}} } } =\mathrm{2}^{\sqrt{\mathrm{2}}} \\ $$$${x}=? \\ $$
Commented by Frix last updated on 11/May/23
$$\mathrm{2}^{{x}^{{x}^{{x}} } } =\mathrm{2}^{\left({x}^{\left({x}^{{x}} \right)} \right)} \\ $$$$\mathrm{ln}\:\mathrm{2}^{{x}^{{x}^{{x}} } } \:=\mathrm{ln}\:\mathrm{2}^{\sqrt{\mathrm{2}}} \\ $$$${x}^{{x}^{{x}} } =\sqrt{\mathrm{2}} \\ $$$${x}^{{x}} \mathrm{ln}\:{x}\:=\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}} \\ $$$$\mathrm{We}\:\mathrm{can}\:\mathrm{only}\:\mathrm{approximate} \\ $$
Commented by mathlove last updated on 11/May/23
$${ok} \\ $$
Answered by josemate19 last updated on 10/May/23
$${ln}\mathrm{2}{x}^{{x}^{{x}} } ={ln}\mathrm{2}^{\sqrt{\mathrm{2}}} \\ $$$${x}^{{x}} {ln}\mathrm{2}{x}=\sqrt{\mathrm{2}}{ln}\mathrm{2} \\ $$$${x}^{{x}} =\frac{\sqrt{\mathrm{2}}{ln}\mathrm{2}}{{ln}\mathrm{2}{x}} \\ $$$${lnx}^{{x}} ={ln}\left(\frac{\sqrt{\mathrm{2}}{ln}\mathrm{2}}{{ln}\mathrm{2}{x}}\right) \\ $$$$ \\ $$$${xlnx}={ln}\left(\frac{\sqrt{\mathrm{2}}{ln}\mathrm{2}}{{ln}\mathrm{2}{x}}\right) \\ $$$$ \\ $$$${x}=\frac{{ln}\left(\frac{\sqrt{\mathrm{2}}{ln}\mathrm{2}}{{ln}\mathrm{2}{x}}\right)}{{lnx}} \\ $$$$ \\ $$$${x}=\frac{{ln}\left(\frac{{ln}\mathrm{2}^{\sqrt{\mathrm{2}}} }{{ln}\mathrm{2}{x}}\right)}{{lnx}} \\ $$
Commented by Frix last updated on 11/May/23
$$\mathrm{Wrong}. \\ $$