Given that ((log a)/(b−c))=((log b)/(c−a))=((log c)/(a−b)) find the value of a^a b^b c^c .


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Question Number 124705 by ZiYangLee last updated on 05/Dec/20

$Given that$ $\frac{\log a}{b - c} = \frac{\log b}{c - a} = \frac{\log c}{a - b}$ $find the value of a^{a} b^{b} c^{c} .$
	Answered by TANMAY PANACEA last updated on 05/Dec/20

	$p = a^{a} b^{b} c^{c}$ $l o g p = a l o g a + b l o g b + c l o g c$ $= a \times k (b - c) + b k (c - a) + c k (a - b)$ $= 0$ $l o g p = 0 = l o g 1$ $p = 1$
	Answered by benjo_mathlover last updated on 05/Dec/20
	${ ((log a^(c−a) = log b^(b−c) )),((log a^(a−b) = log c^(b−c) )) :} { (((a^c /a^a ) = (b^b /b^c ) ⇒(ab)^c = a^a b^b )),(((a^a /a^b ) = (c^b /c^c ) ⇒c^c = (((ac)^b )/a^a ))) :} then a^a b^b c^c = (ab)^c .c^c = (abc)^c$
	${\begin{cases} \log a^{c - a} = \log b^{b - c} \\ \log a^{a - b} = \log c^{b - c} \end{cases}$ ${\begin{cases} \frac{a^{c}}{a^{a}} = \frac{b^{b}}{b^{c}} \Rightarrow {(a b)}^{c} = a^{a} b^{b} \\ \frac{a^{a}}{a^{b}} = \frac{c^{b}}{c^{c}} \Rightarrow c^{c} = \frac{{(a c)}^{b}}{a^{a}} \end{cases}$ $t h e n a^{a} b^{b} c^{c} = {(a b)}^{c} . c^{c} = {(a b c)}^{c}$
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