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For-x-x-x-a-x-a-a-x-a-1-a-Is-this-true-When-is-it-not-true-




Question Number 5591 by FilupSmith last updated on 21/May/16
For x^x^x^(...)   =a  ∴ x^a =a   ⇒   x=a^(1/a)     Is this true?  When is it not true?
$$\mathrm{For}\:{x}^{{x}^{{x}^{…} } } ={a} \\ $$$$\therefore\:{x}^{{a}} ={a}\:\:\:\Rightarrow\:\:\:{x}={a}^{\frac{\mathrm{1}}{{a}}} \\ $$$$ \\ $$$$\mathrm{Is}\:\mathrm{this}\:\mathrm{true}? \\ $$$$\mathrm{When}\:\mathrm{is}\:\mathrm{it}\:\mathrm{not}\:\mathrm{true}? \\ $$
Commented by prakash jain last updated on 21/May/16
If the LHS exists and is finite the   x^a =a.
$$\mathrm{If}\:\mathrm{the}\:\mathrm{LHS}\:\mathrm{exists}\:\mathrm{and}\:\mathrm{is}\:\mathrm{finite}\:\mathrm{the}\: \\ $$$${x}^{{a}} ={a}. \\ $$

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