2^(√x) = x find x


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Question Number 40583 by Cheyboy last updated on 24/Jul/18

$2^{\sqrt{x}} = x$ $f i n d x$
	Answered by $@ty@m last updated on 24/Jul/18

	$4$
	Commented by Cheyboy last updated on 24/Jul/18

	$I n e e d w o r k i n g 16 t o o i s a p o s s i b l e$ $a n s w e r$
	Answered by MrW3 last updated on 24/Jul/18
	$let t=(√x)>0 2^t =t^2 2^(t/2) =t e^((ln 2 t)/2) =t −((ln 2)/2)=−((ln 2 t)/2) e^(−((ln 2 t)/2)) ⇒−((ln 2 t)/2)=W(−((ln 2)/2)) ⇒t=−(2/(ln 2)) W(−((ln 2)/2)) ⇒x=t^2 =[−(2/(ln 2)) W(−((ln 2)/2))]^2 = { (((−((−2×1.3863)/(ln 2)))^2 =16)),(((−((−2×0.6931)/(ln 2)))^2 =4)) :}$
	$l e t t = \sqrt{x} > 0$ $2^{t} = t^{2}$ $2^{\frac{t}{2}} = t$ $e^{\frac{\ln 2 t}{2}} = t$ $- \frac{\ln 2}{2} = - \frac{\ln 2 t}{2} e^{- \frac{\ln 2 t}{2}}$ $\Rightarrow - \frac{\ln 2 t}{2} = W (- \frac{\ln 2}{2})$ $\Rightarrow t = - \frac{2}{\ln 2} W (- \frac{\ln 2}{2})$ $\Rightarrow x = t^{2} = {[- \frac{2}{\ln 2} W (- \frac{\ln 2}{2})]}^{2} = {\begin{cases} {(- \frac{- 2 \times 1.3863}{\ln 2})}^{2} = 16 \\ {(- \frac{- 2 \times 0.6931}{\ln 2})}^{2} = 4 \end{cases}$
	Commented by Cheyboy last updated on 24/Jul/18

	$T h a n k y o u s i r!! R e a l l y a p p r e c i a t e d$
	Commented by MrW3 last updated on 24/Jul/18

	$Y o u a r e w e l c o m e s i r!$ $w h a t i f t h e q u e s t i o n i s :$ $2^{x} = x^{2}$ $f i n d x .$
	Commented by Cheyboy last updated on 25/Jul/18

	$x^{2} = 2^{x}$ $\sqrt{x^{2}} = \sqrt{2^{x}}$ $x = \pm 2^{\frac{x}{2}}$ $x = \pm e^{\frac{x}{2} \ln 2}$ ${x e}^{- \frac{x}{2} \ln 2} = \pm 1$ $- \frac{x}{2} \ln 2 e^{- \frac{x}{2} \ln 2} = - (\pm \frac{\ln 2}{2})$ $x = - \frac{2}{\ln 2} W (\pm \frac{\ln 2}{2})$ $S i r i {c a n}^{^{'}} t c o m p u t e f u r t h e r$ $N o c a l c u l a t o r . .$ $P l z z v e r i f y$
	Commented by MrW3 last updated on 25/Jul/18

	$a b s o l u t e l y c o r r e c t!$ $W (\frac{\ln 2}{2}) h a s o n e v a l u e (0.2657),$ $W (- \frac{\ln 2}{2}) h a s t w o v a l u e s (s e e a b o v e),$ $s o t o t a l l y t h e r e a r e t h r e e s o l u t i o n s$ $f o r x .$
	Commented by Cheyboy last updated on 26/Jul/18

	$T h a n k y o u s i r . . I w i s h y o u c o u l d$ $t e a c h m a t h s a n p h y s i c s . I d o$ $l e a r n a l o t f r o m a n s w e r, a n d$ $o t h e r s . .$
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