x^x =6 x=?


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Question Number 131697 by mathlove last updated on 07/Feb/21

$x^{x} = 6 x = ?$
	Answered by mr W last updated on 07/Feb/21

	$x = 6^{\frac{1}{x}} = e^{\frac{\ln 6}{x}}$ $\frac{1}{x} e^{\frac{\ln 6}{x}} = 1$ $\frac{\ln 6}{x} e^{\frac{\ln 6}{x}} = \ln 6$ $\frac{\ln 6}{x} = W (\ln 6)$ $\Rightarrow x = \frac{\ln 6}{W (\ln 6)} \approx 2.231829$
	Commented by mathlove last updated on 07/Feb/21

	$t h a n k s s i r$
	Commented by mathlove last updated on 07/Feb/21

	$w h a t W (l n 6) = ?$
	Commented by mr W last updated on 07/Feb/21

	$L a m b e r t W f u n c t i o n$ $W (\ln 6) \approx 0.802821260392$
	Answered by Dwaipayan Shikari last updated on 07/Feb/21

	$x^{x} = 6$ $\Rightarrow x l o g (x) = l o g (6) \Rightarrow l o g (x) e^{l o g (x)} = l o g (6) \Rightarrow l o g (x) = W_{0} (l o g (6))$ $x = e^{W_{0} (l o g (6))} = \frac{l o g (6)}{W_{0} (l o g (6))}$ $x = \frac{l o g (6)}{l o g (6) - {l o g}^{2} (6) + \frac{3}{2} {l o g}^{3} (6) - \frac{8}{3} {l o g}^{4} (6) + \frac{125}{24} {l o g}^{5} (6) - . .}$ $E x a c t s o l u t i o n f o r x$ $= \frac{1}{1 - l o g (6) + \frac{3}{2} {l o g}^{2} (6) - \frac{8}{3} {l o g}^{3} (6) + . . .}$
	Commented by I want to learn more last updated on 07/Feb/21

	$Sir, please what is the formular for the W_{0} (\log 6) for the exact form .$
	Commented by Dwaipayan Shikari last updated on 07/Feb/21

	$A s y m t o t i c r e l a t i o n$ $W_{0} (x) = \underset{n = 1}{\sum^{\infty}} \frac{{(- n)}^{n - 1}}{n!} x^{n}$
	Commented by I want to learn more last updated on 07/Feb/21

	$Thanks sir, i appreciate .$
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