solve: x^2 =2^x


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Question Number 172029 by Mikenice last updated on 23/Jun/22

$s o l v e :$ $x^{2} = 2^{x}$
Commented by Rasheed.Sindhi last updated on 23/Jun/22

$x = 2, 4$
Commented by Mikenice last updated on 23/Jun/22

$p l e a s e s h o w t h e s o l u t i o n$
Commented by mr W last updated on 23/Jun/22

$x = - 0.76666 a s w e l l$
Commented by puissant last updated on 23/Jun/22
$x^2 =2^(x ) ⇒ 2lnx = xln2 ⇒ ((lnx)/x) = ((ln2)/2) ((lnx)/x) = ((2ln2)/4) = ((ln4)/4). S={2 ; 4}$
$x^{2} = 2^{x} \Rightarrow 2 l n x = x l n 2$ $\Rightarrow \frac{l n x}{x} = \frac{l n 2}{2}$ $\frac{l n x}{x} = \frac{2 l n 2}{4} = \frac{l n 4}{4} .$ $S = {2; 4}$
Commented by mr W last updated on 23/Jun/22

$f r o m \frac{a}{b} = \frac{c}{d} w e g e t n o t o n l y$ $a = c, b = d .$ $g e n e r a l l y w e g e t$ $a = k c, b = k d .$
	Answered by mr W last updated on 25/Jun/22
	$x^2 =2^x x=±2^(x/2) =±e^((xln 2)/2) (−((xln 2)/2))e^(−((xln 2)/2)) =±((ln 2)/2) −((xln 2)/2)=W(±((ln 2)/2)) ⇒x=−(2/(ln 2))W(±((ln 2)/2)) = { ((−(2/(ln 2))W(−((ln 2)/2))= { (2),(4) :})),((−(2/(ln 2))W(((ln 2)/2))=−0.766665)) :}$
	$x^{2} = 2^{x}$ $x = \pm 2^{\frac{x}{2}} = \pm e^{\frac{x \ln 2}{2}}$ $(- \frac{x \ln 2}{2}) e^{- \frac{x \ln 2}{2}} = \pm \frac{\ln 2}{2}$ $- \frac{x \ln 2}{2} = W (\pm \frac{\ln 2}{2})$ $\Rightarrow x = - \frac{2}{\ln 2} W (\pm \frac{\ln 2}{2})$ $= {\begin{cases} - \frac{2}{\ln 2} W (- \frac{\ln 2}{2}) = {\begin{cases} 2 \\ 4 \end{cases} \\ - \frac{2}{\ln 2} W (\frac{\ln 2}{2}) = - 0.766665 \end{cases}$
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