solve x^2^x =−(1/2)


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Question Number 120780 by mathace last updated on 02/Nov/20

$s o l v e x^{2^{x}} = - \frac{1}{2}$
Commented by Anuragkar last updated on 02/Nov/20

$U s e L a m b e r t W f u n c t i o n t o w o r k i t o u t . . . . . .$
Commented by Dwaipayan Shikari last updated on 02/Nov/20

$C o m p l e x s o l u t i o n (x \in C) . N o r e a l s o l u t i o n$
	Answered by Anuragkar last updated on 14/Nov/20
	$2^x ln(x)=ln(e^(iπ) )−ln(2) xln(x)ln(2)=ln(iπ−ln(2)) ln(x)e^(ln(x)) =ln(iπ−ln(2))/ln(2) Now use lambert W function W(ln(x)e^(ln(x)) )=ln(x)=W[ln{iπ−ln(2)}/ln(2)] x=e^(W[ln{iπ−ln(2)}/ln(2)]) .....this is the final answer....calculstw it using Wolframalpha$
	$2^{x} l n (x) = l n (e^{i π}) - l n (2)$ $x l n (x) l n (2) = l n (i π - l n (2))$ $l n (x) e^{l n (x)} = l n (i π - l n (2)) / l n (2)$ $N o w u s e l a m b e r t W f u n c t i o n$ $W (l n (x) e^{l n (x)}) = l n (x) = W [l n {i π - l n (2)} / l n (2)]$ $x = e^{W [l n {i π - l n (2)} / l n (2)]}$ $. . . . . t h i s i s t h e f i n a l a n s w e r . . . . c a l c u l s t w i t u s i n g$ $W o l f r a m a l p h a$
	Commented by Dwaipayan Shikari last updated on 13/Nov/20

	$E r r o r i n s e c o n d l i n e$
	Answered by mathace last updated on 18/Nov/20

	$x^{2^{x}} = - \frac{1}{2}$ $2^{x} \ln (x) = \ln (e^{i π}) - \ln (2)$ $e^{x \ln (2)} \ln (x) = i π - \ln (2)$ $\ln (e^{x \ln (2)} \ln (x)) = \ln (i π - \ln (2))$ $x \ln (2) + \ln (\ln (x)) = \ln (i π - \ln (2))$ $e^{e^{t}} α + t = β w h e r e α = \ln (2), β = \ln (i π - \ln (2))$ $α + {t e}^{- e^{t}} = β e^{- e^{t}}$ $(t - β) e^{- e^{t}} = - α$ $(t - β) e^{- (1 + t + O (t^{2}))} = - α$ $(t - β) e^{- (1 + t)} = - α$ $(t - β) e^{- t} = \frac{- α}{e^{- 1}} = - α e$ $(β - t) e^{- t} = α e$ $(β - t) e^{- t} e^{β} = α e^{β + 1}$ $(β - t) e^{β - t} = α e^{β + 1}$ $W ((β - t) e^{β - t}) = W (α e^{β + 1})$ $β - t = W (α e^{β + 1})$ $t = β - W (α e^{β + 1})$ $N o w r e c a l l i n g$ $x = e^{e^{t}}$ $x = e^{e^{β - W (α e^{β + 1})}}$ $R e c a l l i n g t h e v a l u e o f α a n d β, w e g e t$ $t h e r e q u i r e d s o l u t i o n .$ $(O m a r)$
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