2-x-x-find-x-

Question Number 40583 by Cheyboy last updated on 24/Jul/18

2^{\sqrt{x}} = x

f i n d x

Answered by $@ty@m last updated on 24/Jul/18

4

Commented by Cheyboy last updated on 24/Jul/18

I n e e d w o r k i n g 16 t o o i s a p o s s i b l e

a n s w e r

Answered by MrW3 last updated on 24/Jul/18

l e t t = \sqrt{x} > 0

2^{t} = t^{2}

2^{\frac{t}{2}} = t

e^{\frac{\ln 2 t}{2}} = t

- \frac{\ln 2}{2} = - \frac{\ln 2 t}{2} e^{- \frac{\ln 2 t}{2}}

\Rightarrow - \frac{\ln 2 t}{2} = W (- \frac{\ln 2}{2})

\Rightarrow t = - \frac{2}{\ln 2} W (- \frac{\ln 2}{2})

\Rightarrow x = t^{2} = {[- \frac{2}{\ln 2} W (- \frac{\ln 2}{2})]}^{2} = {\begin{cases} {(- \frac{- 2 \times 1.3863}{\ln 2})}^{2} = 16 \\ {(- \frac{- 2 \times 0.6931}{\ln 2})}^{2} = 4 \end{cases}

Commented by Cheyboy last updated on 24/Jul/18

T h a n k y o u s i r!! R e a l l y a p p r e c i a t e d

Commented by MrW3 last updated on 24/Jul/18

Y o u a r e w e l c o m e s i r!

w h a t i f t h e q u e s t i o n i s :

2^{x} = x^{2}

f i n d x .

Commented by Cheyboy last updated on 25/Jul/18

x^{2} = 2^{x}

\sqrt{x^{2}} = \sqrt{2^{x}}

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

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

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

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

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

S i r i {c a n}^{^{'}} t c o m p u t e f u r t h e r

N o c a l c u l a t o r . .

P l z z v e r i f y

Commented by MrW3 last updated on 25/Jul/18

a b s o l u t e l y c o r r e c t!

W (\frac{\ln 2}{2}) h a s o n e v a l u e (0.2657),

W (- \frac{\ln 2}{2}) h a s t w o v a l u e s (s e e a b o v e),

s o t o t a l l y t h e r e a r e t h r e e s o l u t i o n s

f o r x .

Commented by Cheyboy last updated on 26/Jul/18

T h a n k y o u s i r . . I w i s h y o u c o u l d

t e a c h m a t h s a n p h y s i c s . I d o

l e a r n a l o t f r o m a n s w e r, a n d

o t h e r s . .

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