x^x = 64 ⇒ x = ?


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Question Number 148395 by mathdanisur last updated on 27/Jul/21

$x^{x} = 64$ $\Rightarrow x = ?$
	Answered by Math_Freak last updated on 27/Jul/21

	$i f x > 0$ $x l n x = l n 64$ $x l n x = l n 2^{6}$ $x l n x = 6 l n 2$ $(e^{l n x}) ∙ l n x = 6 l n 2$ $i n t r o d u c i n g l a m b e r t f u n c t i o n$ $l n x = W (6 l n 2)$ $x = e^{W (6 l n 2)} \approx 3.39121540052283312$ $i f x < 0$ $l e t x = - u$ $x l n x = 6 l n 2$ $- u l n ∣ - u ∣ = 6 l n 2$ $- u l n u = 6 l n 2$ $u l n u = - 6 l n 2$ $(e^{l n u}) ∙ l n u = - 6 l n 2$ $i n t r o d u c i n g p r o d u c t l o g$ $l n u = W (- 6 l n 2)$ $u = e^{W (- 6 l n 2)}$ $x = - u$ $x = - e^{W (- 6 l n 2)} \approx 0.7007 - 1.9045 i$
	Commented by Tawa11 last updated on 27/Jul/21

	$Sir, please how can we get the complex number from - e^{W (- 6 \ln 2)} .$ $How can I compute it . Thanks sir .$
	Commented by Math_Freak last updated on 27/Jul/21

	$U s e w o l f r a m A l p h a$ $s e a r c h f o r w o l f r a m A l p h a o n t h e$ $w e b$ $t o c o m p u t e W (2 l n 2)$ $t y p e p r o d u c t l o g (2 l n 2)$
	Commented by mathdanisur last updated on 27/Jul/21

	$T h a n k y o u S i r c o o l$
	Commented by mathdanisur last updated on 27/Jul/21

	$A n s w e r S i r . ?$
	Commented by mr W last updated on 27/Jul/21

	${(0.7007 - 1.9045 i)}^{(0.7007 - 1.9045 i)} \neq 64$
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