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Question Number 184376 by Ml last updated on 05/Jan/23

2^x =x^2   x=2  x=4  x=−0.76666  ???????????????????  solution please

$$\mathrm{2}^{\mathrm{x}} =\mathrm{x}^{\mathrm{2}} \\ $$$$\mathrm{x}=\mathrm{2} \\ $$$$\mathrm{x}=\mathrm{4} \\ $$$$\mathrm{x}=−\mathrm{0}.\mathrm{76666} \\ $$$$??????????????????? \\ $$$$\mathrm{solution}\:\mathrm{please} \\ $$

Answered by Frix last updated on 05/Jan/23

2^x =x^2   x=−((2t)/(ln 2))  ⇒  (te^t )^2 =(((ln 2)/2))^2   te^t =±((ln 2)/2)  t=W (±((ln 2)/2)) = { ((ln (1/4) ∨ln (1/2))),((≈.265705736221)) :}  ⇒  x= { ((4∨2)),((≈−.766664695962)) :}

$$\mathrm{2}^{{x}} ={x}^{\mathrm{2}} \\ $$$${x}=−\frac{\mathrm{2}{t}}{\mathrm{ln}\:\mathrm{2}} \\ $$$$\Rightarrow \\ $$$$\left({t}\mathrm{e}^{{t}} \right)^{\mathrm{2}} =\left(\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$${t}\mathrm{e}^{{t}} =\pm\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}} \\ $$$${t}={W}\:\left(\pm\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}}\right)\:=\begin{cases}{\mathrm{ln}\:\frac{\mathrm{1}}{\mathrm{4}}\:\vee\mathrm{ln}\:\frac{\mathrm{1}}{\mathrm{2}}}\\{\approx.\mathrm{265705736221}}\end{cases} \\ $$$$\Rightarrow \\ $$$${x}=\begin{cases}{\mathrm{4}\vee\mathrm{2}}\\{\approx−.\mathrm{766664695962}}\end{cases} \\ $$

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