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Question-116347




Question Number 116347 by Khalmohmmad last updated on 03/Oct/20
Answered by MJS_new last updated on 03/Oct/20
x^(−x^x ) =x^((−x^x )) =(1/x^((x^x )) )  ⇒ x^((x^x )) =2^(−(√2))   x^x ln x =−(√2)ln 2  x^x  defined for x>0 ⇒ ln x <0 ⇒ 0<x<1  I think ln x =aln 2 ⇒ x=2^a ∧a<0  trying leads to a=−2 ⇒ x=(1/4)
$${x}^{−{x}^{{x}} } ={x}^{\left(−{x}^{{x}} \right)} =\frac{\mathrm{1}}{{x}^{\left({x}^{{x}} \right)} } \\ $$$$\Rightarrow\:{x}^{\left({x}^{{x}} \right)} =\mathrm{2}^{−\sqrt{\mathrm{2}}} \\ $$$${x}^{{x}} \mathrm{ln}\:{x}\:=−\sqrt{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$${x}^{{x}} \:\mathrm{defined}\:\mathrm{for}\:{x}>\mathrm{0}\:\Rightarrow\:\mathrm{ln}\:{x}\:<\mathrm{0}\:\Rightarrow\:\mathrm{0}<{x}<\mathrm{1} \\ $$$$\mathrm{I}\:\mathrm{think}\:\mathrm{ln}\:{x}\:={a}\mathrm{ln}\:\mathrm{2}\:\Rightarrow\:{x}=\mathrm{2}^{{a}} \wedge{a}<\mathrm{0} \\ $$$$\mathrm{trying}\:\mathrm{leads}\:\mathrm{to}\:{a}=−\mathrm{2}\:\Rightarrow\:{x}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$

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