Question-12365

Question Number 12365 by chux last updated on 20/Apr/17

Commented by mrW1 last updated on 21/Apr/17

D e f i n i t i o n o f L a m b e r t W f u n c t i o n :

i f y = {x e}^{x}

t h e n x = W (y)

i . e . y = W (y) e^{W (y)}

A l l w h a t y o u h a v e t o d o i s t o t r a n s f o r m

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

Y = X e^{X}

t h e n y o u c a n u s e t h e W - f u n c t i o n t o g e t

X = W (Y)

S o m e t i m e s {i t}^{'} s u s e f u l l t o k n o w

a^{b} = e^{\ln a^{b}} = e^{b \ln a}

F o r e x a m p l e : s o l v e \log x^{2} = \frac{x}{8}

\Rightarrow x^{2} = 10^{(\frac{x}{8})}

\Rightarrow x = \pm 10^{(\frac{x}{16})}

\Rightarrow x = \pm e^{(\frac{x}{16}) \ln 10}

\Rightarrow x \times e^{- \frac{\ln 10}{16} x} = \pm 1

\Rightarrow (- \frac{\ln 10}{16} x) \times e^{(- \frac{\ln 10}{16} x)} = \pm \frac{\ln 10}{16}

\Rightarrow - \frac{\ln 10}{16} x = W (\pm \frac{\ln 10}{16})

\Rightarrow x = - \frac{W (\pm \frac{\ln 10}{16})}{\frac{\ln 10}{16}}

\approx - \frac{W (\pm 0.143911)}{0.143911}

= {\begin{cases} - \frac{W (0.143911)}{0.143911} = - \frac{0.126776}{0.143911} = - 0.88093 \\ - \frac{W (- 0.143911)}{0.143911} = {\begin{cases} - \frac{- 0.170697}{0.143911} = 1.18613 \\ - \frac{- 3.05550}{0.143911} = 21.23178 \end{cases} \end{cases}

F o r e x a m p l e : s o l v e x^{x} = 16

\Rightarrow x \ln x = \ln 16

\Rightarrow (\ln x) e^{(\ln x)} = \ln 16

\Rightarrow \ln x = W (\ln 16)

\Rightarrow x = e^{W (\ln 16)} = \frac{\ln 16}{W (\ln 16)}

\approx \frac{2.771588}{W (2.771588)} = \frac{2.771588}{1.00973} = 2.74587

I n i n t e r n e t y o u m a y f i n d c a l c u l a t o r s

t o e v a l u a t e W f u n c t i o n v a l u e s .

Commented by chux last updated on 20/Apr/17

i ve always had problems with the

final simplification using

LAMBART W FUNCTION .

is it that there s a special calculator

for it or what .

please help me out if theres any

calculator or rule for it . Also help

with its properties or rules .

Thanks for always helping .

Commented by chux last updated on 21/Apr/17

can you recomend any calculator

for it .

Commented by mrW1 last updated on 21/Apr/17

T h e r e w a s a v e r y n i c e o n l i n e

c a l c u l a t o r a t w w w . h a d 2 k n o w . c o m,

b u t t h e s i t e i s n o l o n g e r v a l i d .

I u s e g e o g e b r a t o e v a l u a t e W (a)

i n d i r e c t l y .

Commented by chux last updated on 21/Apr/17

thank you very much sir

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