x-x-6-x-

x-x-6-x-

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) + \dots}

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 .