solve the value of x logx^2 =(x/(25))


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Question Number 10248 by j.masanja06@gmail.com last updated on 31/Jan/17

$solve the value of x$ ${logx}^{2} = \frac{x}{25}$
	Answered by mrW1 last updated on 01/Feb/17
	$I. if x>0: log x^2 =(x/(25)) ⇒2log x=(x/(25)) log x=(x/(50)) ((ln x)/(ln 10))=(x/(50)) ln x=((ln 10)/(50))x=−ax with a=−((ln 10)/(50)) ln x=−ax x=e^(−ax) xe^(ax) =1 (ax)e^(ax) =a ⇒ax=W(a) Lambert W function ⇒x=((W(a))/a)=−((W(−((ln 10)/(50))))/((ln 10)/(50)))=−((W(−0.046052))/(0.046052)) { ((=((−0.048332)/(−0.046052))=1.049516)),((=((−4.605162)/(−0.046052))=100)) :} II. if x<0: log x^2 =(x/(25)) let t=−x, t>0 log (−t)^2 =−(t/(25)) log (t)^2 =−(t/(25)) 2log t=−(t/(25)) see above t=((W(((ln 10)/(50))))/((ln 10)/(50))) x=−t=−((W(((ln 10)/(50))))/((ln 10)/(50)))=−((W(0.046052))/(0.046052)) =−((0.044067)/(0.046052))=−0.956903 ⇒ x=−((W(±((ln 10)/(50))))/((ln 10)/(50)))≈ { ((−0.956903)),((1.049516)),((100)) :}$
	$I . i f x > 0 :$ $\log x^{2} = \frac{x}{25}$ $\Rightarrow 2 \log x = \frac{x}{25}$ $\log x = \frac{x}{50}$ $\frac{\ln x}{\ln 10} = \frac{x}{50}$ $\ln x = \frac{\ln 10}{50} x = - a x$ $w i t h a = - \frac{\ln 10}{50}$ $\ln x = - a x$ $x = e^{- a x}$ ${x e}^{a x} = 1$ $(a x) e^{a x} = a$ $\Rightarrow a x = W (a) L a m b e r t W f u n c t i o n$ $\Rightarrow x = \frac{W (a)}{a} = - \frac{W (- \frac{\ln 10}{50})}{\frac{\ln 10}{50}} = - \frac{W (- 0.046052)}{0.046052}$ ${\begin{cases} = \frac{- 0.048332}{- 0.046052} = 1.049516 \\ = \frac{- 4.605162}{- 0.046052} = 100 \end{cases}$ $I I . i f x < 0 :$ $\log x^{2} = \frac{x}{25}$ $l e t t = - x, t > 0$ $\log {(- t)}^{2} = - \frac{t}{25}$ $\log {(t)}^{2} = - \frac{t}{25}$ $2 \log t = - \frac{t}{25}$ $s e e a b o v e$ $t = \frac{W (\frac{\ln 10}{50})}{\frac{\ln 10}{50}}$ $x = - t = - \frac{W (\frac{\ln 10}{50})}{\frac{\ln 10}{50}} = - \frac{W (0.046052)}{0.046052}$ $= - \frac{0.044067}{0.046052} = - 0.956903$ $\Rightarrow x = - \frac{W (\pm \frac{\ln 10}{50})}{\frac{\ln 10}{50}} \approx {\begin{cases} - 0.956903 \\ 1.049516 \\ 100 \end{cases}$
	Answered by arge last updated on 04/Feb/17

	$2 l o g x = \frac{x}{25}$ $l o g x = y$ $x = 50 y$
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