Solve for x : x + ln(1−x) = 0.1614 Help!


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Question Number 191129 by Mastermind last updated on 18/Apr/23

$Solve for x :$ $x + \ln (1 - x) = 0.1614$ $Help!$
	Answered by Skabetix last updated on 18/Apr/23

	$no solution for x \in r e a l s n u m b e r s$
	Answered by mr W last updated on 22/Apr/23
	$f(x)=x+ln (1−x)≤0 so f(x)=0.1614 has no solution! say the question is x+ln (1−x)=−0.1614, then we can solve it as following: x+ln (1−x)=−0.1614 ln e^x (1−x)=−0.1614 e^x (1−x)=e^(−0.1614) e^x (x−1)=−e^(−0.1614) e^(x−1) (x−1)=−e^(−0.1614−1) ⇒x−1=W(−e^(−0.1614−1) ) ⇒x=1+W(−e^(−0.1614−1) ) ≈ { ((−0.68047868)),((0.46604647)) :}$
	$f (x) = x + \ln (1 - x) ⩽ 0$ $s o f (x) = 0.1614 h a s n o s o l u t i o n!$ $s a y t h e q u e s t i o n i s$ $x + \ln (1 - x) = - 0.1614, t h e n w e c a n$ $s o l v e i t a s f o l l o w i n g :$ $x + \ln (1 - x) = - 0.1614$ $\ln e^{x} (1 - x) = - 0.1614$ $e^{x} (1 - x) = e^{- 0.1614}$ $e^{x} (x - 1) = - e^{- 0.1614}$ $e^{x - 1} (x - 1) = - e^{- 0.1614 - 1}$ $\Rightarrow x - 1 = W (- e^{- 0.1614 - 1})$ $\Rightarrow x = 1 + W (- e^{- 0.1614 - 1})$ $\approx {\begin{cases} - 0.68047868 \\ 0.46604647 \end{cases}$
	Commented by Mastermind last updated on 26/Apr/23

	$Oh! like seriously I {did}^{'} nt noticed$ ${you}^{'} ve helped me .$ $Thank you so much$
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