Solve: ((x/4))^(log_5 50x) = x^6


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Question Number 52007 by Tawa1 last updated on 02/Jan/19

$Solve : {(\frac{x}{4})}^{\log_{5} 50 x} = x^{6}$
	Answered by MJS last updated on 02/Jan/19

	${(\frac{x}{4})}^{\log_{5} 50 x} = x^{6}$ $\ln \frac{x}{4} \log_{5} 50 x = 6 \ln x$ $(\ln x - \ln 4) \frac{\ln x + \ln 50}{\ln 5} - 6 \ln x = 0$ ${(\ln x)}^{2} - \ln 1250 \ln x - 2 \ln 50 \ln 2 = 0$ $\ln x = \frac{\ln 1250}{2} \pm \sqrt{\frac{{(\ln 1250)}^{2}}{4} + 2 \ln 50 \ln 2}$ $\ln 1250 = 4 \ln 5 + \ln 2$ ${(\ln 1250)}^{2} = 16 {(\ln 5)}^{2} + 8 \ln 5 \ln 2 + {(\ln 2)}^{2}$ $2 \ln 50 \ln 2 = 4 \ln 5 \ln 2 + 2 {(\ln 2)}^{2}$ $\sqrt{. . .} = \sqrt{4 {(\ln 5)}^{2} + 6 \ln 5 \ln 2 + \frac{9}{4} {(\ln 2)}^{2}} =$ $= \sqrt{\frac{{(4 \ln 5 + 3 \ln 2)}^{2}}{4}} = \frac{4 \ln 5 + 3 \ln 2}{2}$ $\ln x = \frac{4 \ln 5 + \ln 2}{2} \pm \frac{4 \ln 5 + 3 \ln 2}{2}$ $\ln x = - \ln 2 \lor \ln x = 4 \ln 5 + 2 \ln 2$ $\ln x = \ln \frac{1}{2} \lor \ln x = \ln 2500$ $x = \frac{1}{2} \lor x = 2500$
	Commented by Tawa1 last updated on 02/Jan/19

	$God bless you sir$
	Answered by tanmay.chaudhury50@gmail.com last updated on 02/Jan/19

	${l o g}_{5} 50 x \times (l n x - l n 4) = 6 l n x$ ${l o g}_{a} b = \frac{l n b}{l n a} s o {l n}_{5} 50 x = \frac{l n 50 x}{l n 5}$ $(\frac{l n x + l n 50}{l n 5}) \times (l n x - l n 4) = 6 l n x$ $(\frac{l n x + 2 l n 5 + l n 2}{l n 5}) (l n x - 2 l n 2) = 6 l n x$ $(\frac{k + 2 b + a}{b}) (k - 2 a) = 6 k$ $k^{2} + k (2 b + a) - 2 a k - 4 a b - 2 a^{2} = 6 b k$ $k^{2} + k (2 b + a - 2 a - 6 b) - 4 a b - 2 a^{2} = 0$ $k^{2} - k (a + 4 b) - (4 a b + 2 a^{2}) = 0$ $k = \frac{(a + 4 b) \pm \sqrt{{(a + 4 b)}^{2} + 4 \times 1 \times (4 a b + 2 a^{2})}}{2}$ $k = \frac{(a + 4 b) \pm \sqrt{a^{2} + 8 a b + 16 b^{2} + 16 a b + 8 a^{2}}}{2}$ $k = \frac{(a + 4 b) \pm \sqrt{9 a^{2} + 24 a b + 16 b^{2}}}{2}$ $k = \frac{(a + 4 b) \pm \sqrt{{(3 a + 4 b)}^{2}}}{2}$ $k = \frac{a + 4 b + 3 a + 4 b}{2}, \frac{a + 4 b - 3 a - 4 b}{2}$ $k = (2 a + 4 b), (- a)$ $l n x = 2 l n 2 + 4 l n 5$ $= l n (4 \times 5^{4})$ $l n x = l n 2500 s o x = 2500$ $l n x = - a = - l n 2$ $l n x = l n 2^{- 1} s o x = \frac{1}{2}$
	Commented by Tawa1 last updated on 02/Jan/19

	$Yes sir . from where you multiply$ $mistake start after this (\frac{k + 2 b + a}{b}) (k - 2 a) = 6 k$
	Commented by Tawa1 last updated on 02/Jan/19

	$God bless you sir$
	Commented by Tawa1 last updated on 02/Jan/19

	$God bless you sir$
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