x-2-2-x-x-

Question Number 80733 by key of knowledge last updated on 05/Feb/20

x^{2} = 2^{x} \Rightarrow x = ?

Commented by mr W last updated on 05/Feb/20

x^{2} = 2^{x}

x = \pm 2^{\frac{x}{2}} = \pm e^{\frac{x \ln 2}{2}}

{x e}^{- \frac{x \ln 2}{2}} = \pm 1

(- \frac{x \ln 2}{2}) e^{(- \frac{x \ln 2}{2})} = \pm \frac{\ln 2}{2}

(- \frac{x \ln 2}{2}) = W (\pm \frac{\ln 2}{2})

x = - \frac{2}{\ln 2} W (\pm \frac{\ln 2}{2}) = {\begin{cases} - \frac{2}{\ln} W (\frac{\ln 2}{2}) = - 0.766665 \\ - \frac{2}{\ln 2} W (- \frac{\ln 2}{2}) = {\begin{cases} - \frac{2 \times (- 0.693147)}{\ln 2} = 2 \\ - \frac{2 \times (- 1.386294)}{\ln 2} = 4 \end{cases} \end{cases}

Commented by key of knowledge last updated on 06/Feb/20

mr \bar{w}, what meaning of W (\mp \frac{\ln 2}{2})

Commented by mr W last updated on 06/Feb/20

{t h a t}^{'} s L a m b e r t W f u n c t i o n, t h e

i n v e r s e f u n c t i o n f r o m y = {x e}^{x} .

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