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Question Number 110725 by Study last updated on 30/Aug/20

x^x^x^x^(...)    =2       x=?

$${x}^{{x}^{{x}^{{x}^{...} } } } =\mathrm{2}\:\:\:\:\:\:\:{x}=? \\ $$

Answered by Her_Majesty last updated on 30/Aug/20

x^x^x^(...)   =2  lnx^x^x^(...)   =ln2  x^x^x^(...)   lnx=ln2  but x^x^x^(...)   =2 ⇒  2lnx=ln2  lnx=((ln2)/2)  x=(√2)

$${x}^{{x}^{{x}^{...} } } =\mathrm{2} \\ $$$${lnx}^{{x}^{{x}^{...} } } ={ln}\mathrm{2} \\ $$$${x}^{{x}^{{x}^{...} } } {lnx}={ln}\mathrm{2} \\ $$$${but}\:{x}^{{x}^{{x}^{...} } } =\mathrm{2}\:\Rightarrow \\ $$$$\mathrm{2}{lnx}={ln}\mathrm{2} \\ $$$${lnx}=\frac{{ln}\mathrm{2}}{\mathrm{2}} \\ $$$${x}=\sqrt{\mathrm{2}} \\ $$

Answered by Dwaipayan Shikari last updated on 30/Aug/20

x^x^(xx^x^x^   )  =2  x^2 =2  x=(√2)

$${x}^{{x}^{{xx}^{{x}^{{x}^{} } } } } =\mathrm{2} \\ $$$${x}^{\mathrm{2}} =\mathrm{2} \\ $$$${x}=\sqrt{\mathrm{2}} \\ $$$$ \\ $$$$ \\ $$

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