x^n =n^x x=?


Question and Answers Forum

All Questions Topic List
Number Theory Questions
Previous in All Question Next in All Question
Previous in Number Theory Next in Number Theory
Question Number 164279 by amin96 last updated on 15/Jan/22

$x^{n} = n^{x} x = ?$
	Answered by Saiki last updated on 15/Jan/22

	${l n x}^{n} = {l n n}^{x}$ $n l n x = x l n n$ $\frac{l n x}{x} = \frac{l n n}{n}$ $x^{- 1} l n x = \frac{l n n}{n}$ $e^{{l n x}^{- 1}} l n x = \frac{l n n}{n}$ $e^{- l n x} l n x = \frac{l n n}{n}$ $- {l n x e}^{- l n x} = - \frac{l n n}{n}$ $W (- {l n x e}^{- l n x}) = W (- \frac{l n n}{n})$ $- l n x = W (- \frac{l n n}{n})$ $x = e^{- W (- \frac{l n n}{n})}$ $s o l u t i o n b y C A S I O . . . . . .$
	Answered by MathsFan last updated on 15/Jan/22

	$x ∙ n^{- \frac{x}{n}} = 1$ $x l n n ∙ e^{- \frac{x}{n} l n n} = l n n$ $- \frac{x l n n}{n} ∙ e^{- \frac{x}{n} l n n} = - \frac{l n n}{n}$ $- \frac{x l n n}{n} = W (- \frac{l n n}{n})$ $x = - \frac{n W (- \frac{l n n}{n})}{l n n}$
	Answered by mr W last updated on 15/Jan/22

	$i f n = e v e n i n t e g e r : e . g . x^{2} = 2^{x}$ $x = \pm n^{\frac{x}{n}}$ $x = \pm e^{\frac{x}{n} \ln n}$ $(- \frac{x}{n} \ln n) e^{- \frac{x}{n} \ln n} = \pm \frac{\ln n}{n}$ $(- \frac{x}{n} \ln n) = W (\pm \frac{\ln n}{n})$ $\Rightarrow x = - \frac{n}{\ln n} W (\pm \frac{\ln n}{n})$ $s i m i l a r l y, i f n \neq e v e n i n t e g e r, e . g . x^{3} = 3^{x} :$ $\Rightarrow x = - \frac{n}{\ln n} W (- \frac{\ln n}{n})$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum