Question Number 209474 by Ismoiljon_008 last updated on 10/Jul/24
Commented by Ismoiljon_008 last updated on 10/Jul/24
$${help}\:{me}\:{please} \\ $$
Answered by Frix last updated on 10/Jul/24
$${t}^{{t}} =\mathrm{2}^{\mathrm{10}\sqrt{\mathrm{2}}} \\ $$$${t}=\mathrm{2}^{{u}} \\ $$$$\left(\mathrm{2}^{{u}} \right)^{\mathrm{2}^{{u}} } =\mathrm{2}^{\mathrm{10}\sqrt{\mathrm{2}}} \\ $$$$\mathrm{2}^{{u}} \mathrm{ln}\:\mathrm{2}^{{u}} \:=\mathrm{10}\sqrt{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$$\mathrm{2}^{{u}} {u}=\mathrm{10}\sqrt{\mathrm{2}}=\mathrm{10}×\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{5}×\mathrm{2}^{\frac{\mathrm{3}}{\mathrm{2}}} =\frac{\mathrm{5}}{\mathrm{2}}×\mathrm{2}^{\frac{\mathrm{5}}{\mathrm{2}}} \\ $$$${u}=\frac{\mathrm{5}}{\mathrm{2}} \\ $$$${t}=\mathrm{2}^{\frac{\mathrm{5}}{\mathrm{2}}} =\mathrm{2}^{\mathrm{2}+\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{4}\sqrt{\mathrm{2}} \\ $$
Commented by Ismoiljon_008 last updated on 11/Jul/24
$${thanks} \\ $$
Answered by Kademi last updated on 11/Jul/24
$$\: \\ $$$$\:\:\:\:\:{t}^{{t}} \:=\:\mathrm{2}^{\mathrm{10}\sqrt{\mathrm{2}}} \\ $$$$\:\:\:\:\:{t}\mathrm{ln}\:{t}\:=\:\mathrm{10}\sqrt{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$$\:\:\:\:\:{W}\left(\mathrm{ln}\:{te}^{\mathrm{ln}\:{t}} \right)\:=\:{W}\left(\frac{\mathrm{5}}{\mathrm{2}}\mathrm{ln}\:\mathrm{2}{e}^{\frac{\mathrm{5}}{\mathrm{2}}\mathrm{ln}\:\mathrm{2}} \:\right) \\ $$$$\:\:\:\:\:\mathrm{ln}\:{t}\:=\:\frac{\mathrm{5ln}\:\mathrm{2}}{\mathrm{2}} \\ $$$$\:\:\:\:\:{t}\:=\:{e}^{\frac{\mathrm{5ln}\:\mathrm{2}}{\mathrm{2}}} \\ $$$$\:\:\:\:\:{t}\:=\:\mathrm{4}\sqrt{\mathrm{2}} \\ $$