x-n-n-x-x-

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 \dots \dots

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})

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