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Question Number 192171 by mathlove last updated on 10/May/23

2^x^x^x =2^(√2) x=?

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

Commented by Frix last updated on 11/May/23

2^x^x^x =2^((x^((x^x )) )) ln 2^x^x^x =ln 2^(√2) x^x^x =(√2) x^x ln x =((ln 2)/2) We can only approximate

$$\mathrm{2}^{{x}^{{x}^{{x}} } } =\mathrm{2}^{\left({x}^{\left({x}^{{x}} \right)} \right)} \\ $$$$\mathrm{ln}\:\mathrm{2}^{{x}^{{x}^{{x}} } } \:=\mathrm{ln}\:\mathrm{2}^{\sqrt{\mathrm{2}}} \\ $$$${x}^{{x}^{{x}} } =\sqrt{\mathrm{2}} \\ $$$${x}^{{x}} \mathrm{ln}\:{x}\:=\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}} \\ $$$$\mathrm{We}\:\mathrm{can}\:\mathrm{only}\:\mathrm{approximate} \\ $$

Commented by mathlove last updated on 11/May/23

ok

$${ok} \\ $$

Answered by josemate19 last updated on 10/May/23

ln2x^x^x =ln2^(√2) x^x ln2x=(√2)ln2 x^x =(((√2)ln2)/(ln2x)) lnx^x =ln((((√2)ln2)/(ln2x))) xlnx=ln((((√2)ln2)/(ln2x))) x=((ln((((√2)ln2)/(ln2x))))/(lnx)) x=((ln(((ln2^(√2) )/(ln2x))))/(lnx))

$${ln}\mathrm{2}{x}^{{x}^{{x}} } ={ln}\mathrm{2}^{\sqrt{\mathrm{2}}} \\ $$$${x}^{{x}} {ln}\mathrm{2}{x}=\sqrt{\mathrm{2}}{ln}\mathrm{2} \\ $$$${x}^{{x}} =\frac{\sqrt{\mathrm{2}}{ln}\mathrm{2}}{{ln}\mathrm{2}{x}} \\ $$$${lnx}^{{x}} ={ln}\left(\frac{\sqrt{\mathrm{2}}{ln}\mathrm{2}}{{ln}\mathrm{2}{x}}\right) \\ $$$$ \\ $$$${xlnx}={ln}\left(\frac{\sqrt{\mathrm{2}}{ln}\mathrm{2}}{{ln}\mathrm{2}{x}}\right) \\ $$$$ \\ $$$${x}=\frac{{ln}\left(\frac{\sqrt{\mathrm{2}}{ln}\mathrm{2}}{{ln}\mathrm{2}{x}}\right)}{{lnx}} \\ $$$$ \\ $$$${x}=\frac{{ln}\left(\frac{{ln}\mathrm{2}^{\sqrt{\mathrm{2}}} }{{ln}\mathrm{2}{x}}\right)}{{lnx}} \\ $$

Commented by Frix last updated on 11/May/23

Wrong.

$$\mathrm{Wrong}. \\ $$

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