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Question Number 50972 by Necxx last updated on 22/Dec/18

	Answered by mr W last updated on 22/Dec/18
	$2^(2x) =8x e^(2xln 2) =8x ((ln 2)/4)e^(2xln 2) =2xln 2 ((ln 2)/4)=(2xln 2)e^(−2xln 2) −((ln 2)/4)=(−2xln 2)e^(−2xln 2) ⇒−2xln 2=W(−((ln 2)/4)) ⇒Lambert function ⇒x=−(1/(2ln 2))W(−((ln 2)/4))= { ((−((−0.2148)/(2 ln 2))=0.1549)),((−((−2.7726)/(2 ln 2))=2)) :}$
	$2^{2 x} = 8 x$ $e^{2 x \ln 2} = 8 x$ $\frac{\ln 2}{4} e^{2 x \ln 2} = 2 x \ln 2$ $\frac{\ln 2}{4} = (2 x \ln 2) e^{- 2 x \ln 2}$ $- \frac{\ln 2}{4} = (- 2 x \ln 2) e^{- 2 x \ln 2}$ $\Rightarrow - 2 x \ln 2 = W (- \frac{\ln 2}{4}) \Rightarrow L a m b e r t f u n c t i o n$ $\Rightarrow x = - \frac{1}{2 \ln 2} W (- \frac{\ln 2}{4}) = {\begin{cases} - \frac{- 0.2148}{2 \ln 2} = 0.1549 \\ - \frac{- 2.7726}{2 \ln 2} = 2 \end{cases}$
	Commented by Necxx last updated on 23/Dec/18

	$t h a n k y o u s i r$
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