Question Number 110725 by Study last updated on 30/Aug/20
$${x}^{{x}^{{x}^{{x}^{...} } } } =\mathrm{2}\:\:\:\:\:\:\:{x}=? \\ $$
Answered by Her_Majesty last updated on 30/Aug/20
$${x}^{{x}^{{x}^{...} } } =\mathrm{2} \\ $$$${lnx}^{{x}^{{x}^{...} } } ={ln}\mathrm{2} \\ $$$${x}^{{x}^{{x}^{...} } } {lnx}={ln}\mathrm{2} \\ $$$${but}\:{x}^{{x}^{{x}^{...} } } =\mathrm{2}\:\Rightarrow \\ $$$$\mathrm{2}{lnx}={ln}\mathrm{2} \\ $$$${lnx}=\frac{{ln}\mathrm{2}}{\mathrm{2}} \\ $$$${x}=\sqrt{\mathrm{2}} \\ $$
Answered by Dwaipayan Shikari last updated on 30/Aug/20
$${x}^{{x}^{{xx}^{{x}^{{x}^{} } } } } =\mathrm{2} \\ $$$${x}^{\mathrm{2}} =\mathrm{2} \\ $$$${x}=\sqrt{\mathrm{2}} \\ $$$$ \\ $$$$ \\ $$