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Question Number 143438 by Rankut last updated on 14/Jun/21

$${\left(\frac{1}{2}\right)}^{\mathit{x}}={\mathit{l}\mathit{o}\mathit{g}}_{\frac{1}{2}}\mathit{x}$$$$\mathit{f}\mathit{i}\mathit{n}\mathit{d}\mathit{x}$$

Answered by mr W last updated on 14/Jun/21

$$x={\left(\frac{1}{2}\right)}^{{\left(\frac{1}{2}\right)}^x}$$$$x={\left(\frac{1}{2}\right)}^x=e^{-x\ln 2}$$$${xe}^{x\ln 2}=1$$$$x\ln 2e^{x\ln 2}=\ln 2$$$$\Rightarrow x=\frac{W\left(\ln 2\right)}{\ln 2}\approx \frac{0.444439091}{\ln 2}=0.64119$$

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