solve using LambertW function ((8/7))^x +17=25x


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Question Number 120154 by mathace last updated on 29/Oct/20

$s o l v e u s i n g L a m b e r t W f u n c t i o n$ ${(\frac{8}{7})}^{x} + 17 = 25 x$
	Answered by mr W last updated on 29/Oct/20
	$((8/7))^x =25(x−((17)/(25))) ((8/7))^(x−((17)/(25))) =25((8/7))^(−((17)/(25))) (x−((17)/(25))) e^((x−((17)/(25)))ln (8/7)) =25((8/7))^(−((17)/(25))) (x−((17)/(25))) −(1/(25))((8/7))^((17)/(25)) ln (8/7)=(((17)/(25))−x)(ln (8/7))e^((((17)/(25))−x)ln (8/7)) W[−(1/(25))((8/7))^((17)/(25)) ln (8/7)]=(((17)/(25))−x)ln (8/7) ⇒x=((17)/(25))−(1/(ln (8/7)))W[−(1/(25))((8/7))^((17)/(25)) ln (8/7)] ≈ { ((0.724057)),((53.864856)) :}$
	${(\frac{8}{7})}^{x} = 25 (x - \frac{17}{25})$ ${(\frac{8}{7})}^{x - \frac{17}{25}} = 25 {(\frac{8}{7})}^{- \frac{17}{25}} (x - \frac{17}{25})$ $e^{(x - \frac{17}{25}) \ln \frac{8}{7}} = 25 {(\frac{8}{7})}^{- \frac{17}{25}} (x - \frac{17}{25})$ $- \frac{1}{25} {(\frac{8}{7})}^{\frac{17}{25}} \ln \frac{8}{7} = (\frac{17}{25} - x) (\ln \frac{8}{7}) e^{(\frac{17}{25} - x) \ln \frac{8}{7}}$ $W [- \frac{1}{25} {(\frac{8}{7})}^{\frac{17}{25}} \ln \frac{8}{7}] = (\frac{17}{25} - x) \ln \frac{8}{7}$ $\Rightarrow x = \frac{17}{25} - \frac{1}{\ln \frac{8}{7}} W [- \frac{1}{25} {(\frac{8}{7})}^{\frac{17}{25}} \ln \frac{8}{7}]$ $\approx {\begin{cases} 0.724057 \\ 53.864856 \end{cases}$
	Commented by mathace last updated on 30/Oct/20

	$C o r r e c t! T h a n k y o u s i r .$
	Commented by mathace last updated on 30/Oct/20

	$W o u l d y o u p l z h e l p t o s o l v e t h e f o l l o w i n g$ $b y L a m b e r t W o r o t h e r s p a c i a l f u n c t i o n$ ${(\frac{8}{7})}^{x} + 17^{x} = 25 x$
	Commented by mr W last updated on 30/Oct/20

	$n o c h a n c e!$
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