x^x =2 find the value of x.


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

$x^{x} = 2$ $f i n d t h e v a l u e o f x .$
	Answered by MrW3 last updated on 12/Jun/18

	$x^{x} = 2$ $x = 2^{\frac{1}{x}}$ $x = e^{\frac{\ln 2}{x}}$ $\ln 2 = \frac{\ln 2}{x} e^{\frac{\ln 2}{x}}$ $\Rightarrow \frac{\ln 2}{x} = W (\ln 2)$ $\Rightarrow x = \frac{\ln 2}{W (\ln 2)} \approx \frac{\ln 2}{0.4444} = 1.5596$ $i n g e n e r a l, t h e s o l u t i o n o f e q n .$ $x^{x} = a w i t h a ⩾ \frac{1}{^{e} \sqrt{e}} \approx 0.6922$ $i s x = \frac{\ln a}{W (\ln a)}$
	Commented by tanmay.chaudhury50@gmail.com last updated on 13/Jun/18

	$w h a t i s t h e m e a n i n g o f W (l n 2) p l s e x p l a i n$
	Commented by MrW3 last updated on 13/Jun/18

	$W (x) i s t h e L a m b e r t W f u n c t i o n w h i c h$ $i s t h e i n v e r s e f u n c t i o n o f y = {x e}^{x} .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 13/Jun/18

	$i h a v e c o l l e c t e d i n f o r m a t i o n a s s i r M R W 3 g u i d e d$ $h e r e i a m p o s t i n g . . . .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 13/Jun/18

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

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

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

	Commented by Rasheed.Sindhi last updated on 14/Jun/18

	$H e l p f u l! E a s y a p p r o a c h!$
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