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Question Number 207986 by Thomaseinstein last updated on 02/Jun/24
(x−3)^(√(x−3 ))   = 3
$$\left({x}−\mathrm{3}\right)^{\sqrt{{x}−\mathrm{3}\:}} \:\:=\:\mathrm{3} \\ $$
Answered by Frix last updated on 02/Jun/24
x−3>0  (√(x−3)) ln (x−3) =3  x=3+e^(2t)   2tln t=ln 3  te^t =((ln 3)/2)  t=W (((ln 3)/2))  x=3+e^(2W (((ln 3)/2)))  ≈5.12479938302
$${x}−\mathrm{3}>\mathrm{0} \\ $$$$\sqrt{{x}−\mathrm{3}}\:\mathrm{ln}\:\left({x}−\mathrm{3}\right)\:=\mathrm{3} \\ $$$${x}=\mathrm{3}+\mathrm{e}^{\mathrm{2}{t}} \\ $$$$\mathrm{2}{t}\mathrm{ln}\:{t}=\mathrm{ln}\:\mathrm{3} \\ $$$${t}\mathrm{e}^{{t}} =\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{2}} \\ $$$${t}=\mathrm{W}\:\left(\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{2}}\right) \\ $$$${x}=\mathrm{3}+\mathrm{e}^{\mathrm{2W}\:\left(\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{2}}\right)} \:\approx\mathrm{5}.\mathrm{12479938302} \\ $$
Commented by Thomaseinstein last updated on 05/Jun/24
this one hmm
$${this}\:{one}\:{hmm} \\ $$

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