solve 2^(x−2) =8x


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Question Number 41390 by mondodotto@gmail.com last updated on 06/Aug/18

$solve 2^{x - 2} = 8 x$
	Answered by MrW3 last updated on 06/Aug/18
	$(2^x /2^2 )=8x 2^x =32x e^(x ln 2) =32x 32x e^(−x ln 2) =1 (−x ln 2) e^(−x ln 2) =−((ln 2)/(32)) ⇒−x ln 2=W(−((ln 2)/(32))) ⇒x=−((W(−((ln 2)/(32))))/(ln 2))= { ((−((−0.0221)/(ln 2))=0.0319)),((−((−5.5452)/(ln 2))=8)) :}$
	$\frac{2^{x}}{2^{2}} = 8 x$ $2^{x} = 32 x$ $e^{x \ln 2} = 32 x$ $32 x e^{- x \ln 2} = 1$ $(- x \ln 2) e^{- x \ln 2} = - \frac{\ln 2}{32}$ $\Rightarrow - x \ln 2 = W (- \frac{\ln 2}{32})$ $\Rightarrow x = - \frac{W (- \frac{\ln 2}{32})}{\ln 2} = {\begin{cases} - \frac{- 0.0221}{\ln 2} = 0.0319 \\ - \frac{- 5.5452}{\ln 2} = 8 \end{cases}$
	Commented by mondodotto@gmail.com last updated on 06/Aug/18

	$thank you sir, but what does W stand for$
	Commented by $@ty@m last updated on 06/Aug/18
	$Lambert W function. If f(x)=xe^x then x=W{f(x)} i.e. W≡f^(−1)$
	$L a m b e r t W f u n c t i o n .$ $I f f (x) = {x e}^{x}$ $t h e n x = W {f (x)}$ $i . e . W \equiv f^{- 1}$
	Commented by mondodotto@gmail.com last updated on 06/Aug/18

	$thanx$
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