Question Number 207986 by Thomaseinstein last updated on 02/Jun/24
$$\left({x}−\mathrm{3}\right)^{\sqrt{{x}−\mathrm{3}\:}} \:\:=\:\mathrm{3} \\ $$
Answered by Frix last updated on 02/Jun/24
$${x}−\mathrm{3}>\mathrm{0} \\ $$$$\sqrt{{x}−\mathrm{3}}\:\mathrm{ln}\:\left({x}−\mathrm{3}\right)\:=\mathrm{3} \\ $$$${x}=\mathrm{3}+\mathrm{e}^{\mathrm{2}{t}} \\ $$$$\mathrm{2}{t}\mathrm{ln}\:{t}=\mathrm{ln}\:\mathrm{3} \\ $$$${t}\mathrm{e}^{{t}} =\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{2}} \\ $$$${t}=\mathrm{W}\:\left(\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{2}}\right) \\ $$$${x}=\mathrm{3}+\mathrm{e}^{\mathrm{2W}\:\left(\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{2}}\right)} \:\approx\mathrm{5}.\mathrm{12479938302} \\ $$
Commented by Thomaseinstein last updated on 05/Jun/24
$${this}\:{one}\:{hmm} \\ $$