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Question Number 172259 by mr W last updated on 25/Jun/22

$$solve$$$$x^x=3^{\frac{1}{x}}$$

Answered by mr W last updated on 25/Jun/22

$$x\ln x=\frac{\ln 3}{x}$$$$x^2\left(2\ln x\right)=2\ln 3$$$$\left(\ln x^2\right)e^{\ln x^2}=2\ln 3$$$$\ln x^2=W\left(2\ln 3\right)$$$$x^2=e^{W\left(2\ln 3\right)}=\frac{2\ln 3}{W\left(2\ln 3\right)}$$$$\Rightarrow x=\sqrt{\frac{2\ln 3}{W\left(2\ln 3\right)}}\approx 1.565552$$

Commented by Tawa11 last updated on 25/Jun/22

$$\mathrm{Great}\mathrm{sir}.$$

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