x-x-2-find-the-value-of-x-

Question Number 37414 by hari321 last updated on 12/Jun/18

$${x}^{{x}} =\mathrm{2} \\ $$$$ \\ $$$${find}\:{the}\:{value}\:{of}\:{x}. \\ $$

Answered by MrW3 last updated on 12/Jun/18

x^x =2 x=2^(1/x) x=e^((ln 2)/x) ln 2=((ln 2)/x)e^((ln 2)/x) ⇒((ln 2)/x)=W(ln 2) ⇒x=((ln 2)/(W(ln 2)))≈((ln 2)/(0.4444))=1.5596 in general, the solution of eqn. x^x =a with a≥(1/(^e (√e)))≈0.6922 is x=((ln a)/(W(ln a)))

$${x}^{{x}} =\mathrm{2} \\ $$$${x}=\mathrm{2}^{\frac{\mathrm{1}}{{x}}} \\ $$$${x}={e}^{\frac{\mathrm{ln}\:\mathrm{2}}{{x}}} \\ $$$$\mathrm{ln}\:\mathrm{2}=\frac{\mathrm{ln}\:\mathrm{2}}{{x}}{e}^{\frac{\mathrm{ln}\:\mathrm{2}}{{x}}} \\ $$$$\Rightarrow\frac{\mathrm{ln}\:\mathrm{2}}{{x}}={W}\left(\mathrm{ln}\:\mathrm{2}\right) \\ $$$$\Rightarrow{x}=\frac{\mathrm{ln}\:\mathrm{2}}{{W}\left(\mathrm{ln}\:\mathrm{2}\right)}\approx\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{0}.\mathrm{4444}}=\mathrm{1}.\mathrm{5596} \\ $$$$ \\ $$$${in}\:{general},\:{the}\:{solution}\:{of}\:{eqn}. \\ $$$${x}^{{x}} ={a}\:{with}\:{a}\geqslant\frac{\mathrm{1}}{\:^{{e}} \sqrt{{e}}}\approx\mathrm{0}.\mathrm{6922} \\ $$$${is}\:{x}=\frac{\mathrm{ln}\:{a}}{{W}\left(\mathrm{ln}\:{a}\right)} \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 13/Jun/18