solve-for-x-10-x-x-50-

Question Number 45314 by mondodotto@gmail.com last updated on 11/Oct/18

solve for x

10^{x} = x^{50}

Answered by MrW3 last updated on 12/Oct/18

10^{x} = x^{50}

\pm 10^{\frac{x}{50}} = x

\pm e^{\frac{x}{50} \ln 10} = x

{x e}^{- \frac{x}{50} \ln 10} = \pm 1

- \frac{x \ln 10}{50} e^{- \frac{x}{50} \ln 10} = \mp \frac{\ln 10}{50}

\Rightarrow - \frac{x \ln 10}{50} = W (\mp \frac{\ln 10}{50})

\Rightarrow x = - \frac{50}{\ln 10} W (\mp \frac{\ln 10}{50})

= {\begin{cases} - \frac{50}{\ln 10} W (\frac{\ln 10}{50}) = - \frac{50 \times 0.044066}{\ln 10} = - 0.956881 \\ - \frac{50}{\ln 10} W (- \frac{\ln 10}{50}) = {\begin{cases} - \frac{50 \times (- 0.0483321)}{\ln 10} = 1.049518 \\ - \frac{50 \times (- 4.605170)}{\ln 10} = 100 \end{cases} \end{cases}

Commented by mondodotto@gmail.com last updated on 12/Oct/18

thanx atleast i get something

Commented by mondodotto@gmail.com last updated on 12/Oct/18

thanx a lot but i dont undstnd this method please can you explain sir

Commented by MrW3 last updated on 12/Oct/18

L a m b e r t W f u n c t i o n i s u s e d .

I f {x e}^{x} = a t h e n x = W (a) .

Commented by mondodotto@gmail.com last updated on 12/Oct/18

what is W

Commented by MrW3 last updated on 12/Oct/18

T o a p p l y L a m b e r t f u n c t i o n t o s o l v e

t h e e q u a t i o n, a l l w h a t w e h a v e t o d o

i s t o p u t t h e e q u a t i o n i n t o a f o r m l i k e

{X e}^{X} = Y

t h e n w e g e t X = W (Y) .

Commented by MrW3 last updated on 12/Oct/18

W (x) i s t h e l a m b e r t f u n c t i o n, d e f i n e d

a s W (x) e^{W (x)} = x .

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