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Question Number 69268 by A8;15: last updated on 22/Sep/19

Answered by mr W last updated on 22/Sep/19

$$x^{x^{20}}=2^{\frac{1}{\sqrt{2}}}$$$$x^x=2^{\frac{1}{20\sqrt{2}}}=a$$$$x=a^{\frac{1}{x}}=e^{\frac{\ln a}{x}}$$$$1=\frac{1}{x}{e}^{\frac{\ln a}{x}}$$$$\ln a=\frac{\ln a}{x}{e}^{\frac{\ln a}{x}}$$$$\Rightarrow \frac{\ln a}{x}=W\left(\ln a\right)$$$$\Rightarrow x=\frac{\ln a}{W\left(\ln a\right)}$$$$\ln a=\frac{\ln 2}{20\sqrt{2}}$$$$\Rightarrow x=\frac{\frac{\ln 2}{20\sqrt{2}}}{W\left(\frac{\ln 2}{20\sqrt{2}}\right)}=\frac{0.02450645}{0.02392705}=1.0242154$$

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