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if-x-x-2-what-is-the-value-of-x-




Question Number 26269 by ibraheem160 last updated on 23/Dec/17
if x^x =2 what is the value of x?
$${if}\:{x}^{{x}} =\mathrm{2}\:{what}\:{is}\:{the}\:{value}\:{of}\:{x}? \\ $$
Answered by mrW1 last updated on 23/Dec/17
x^x =2  xln x=ln 2  e^(ln x) ln x=ln 2  ⇒ln x=W(ln 2)  ⇒x=e^(W(ln 2)) =((ln 2)/(W(ln 2)))  =((0.693147)/(W(0.693147)))=((0.693147)/(0.444446))=1.559610
$${x}^{{x}} =\mathrm{2} \\ $$$${x}\mathrm{ln}\:{x}=\mathrm{ln}\:\mathrm{2} \\ $$$${e}^{\mathrm{ln}\:{x}} \mathrm{ln}\:{x}=\mathrm{ln}\:\mathrm{2} \\ $$$$\Rightarrow\mathrm{ln}\:{x}={W}\left(\mathrm{ln}\:\mathrm{2}\right) \\ $$$$\Rightarrow{x}={e}^{{W}\left(\mathrm{ln}\:\mathrm{2}\right)} =\frac{\mathrm{ln}\:\mathrm{2}}{{W}\left(\mathrm{ln}\:\mathrm{2}\right)} \\ $$$$=\frac{\mathrm{0}.\mathrm{693147}}{{W}\left(\mathrm{0}.\mathrm{693147}\right)}=\frac{\mathrm{0}.\mathrm{693147}}{\mathrm{0}.\mathrm{444446}}=\mathrm{1}.\mathrm{559610} \\ $$

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