Q : ((ln(a ))/(c−b)) =((ln(b))/(a−c)) = ((ln(c))/(b−a)) ⇒ a^( a) . b^( b) . c^( c) =?


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Question Number 173687 by mnjuly1970 last updated on 16/Jul/22

$Q : \frac{l n (a)}{c - b} = \frac{l n (b)}{a - c} = \frac{l n (c)}{b - a}$ $\Rightarrow a^{a} . b^{b} . c^{c} = ?$
	Answered by Rasheed.Sindhi last updated on 17/Jul/22

	$\frac{l n (a)}{c - b} = \frac{l n (b)}{a - c} = \frac{l n (c)}{b - a}$ $\frac{l n (a)}{c - b} = \frac{l n (b)}{a - c}$ $(a - c) \ln a = (c - b) \ln b$ $\ln a^{a - c} = \ln b^{c - b}$ $a^{a - c} = b^{c - b}$ $\frac{a^{a}}{a^{c}} = \frac{b^{c}}{b^{b}}$ $a^{a} b^{b} = {(a b)}^{c} . . . . . . . . . . . . . . . . . . . (i)$ $\frac{l n (b)}{a - c} = \frac{l n (c)}{b - a}$ $(b - a) \ln b = (a - c) \ln c$ $\ln b^{b - a} = \ln c^{a - c}$ $b^{b - a} = c^{a - c}$ $\frac{b^{b}}{b^{a}} = \frac{c^{a}}{c^{c}}$ $b^{b} c^{c} = {(b c)}^{a} . . . . . . . . . . . . . . . . . (i i)$ $\frac{l n (a)}{c - b} = \frac{l n (c)}{b - a}$ $(b - a) \ln a = (c - b) \ln c$ $\ln a^{b - a} = \ln c^{c - b}$ $a^{b - a} = c^{c - b}$ $\frac{a^{b}}{a^{a}} = \frac{c^{c}}{c^{b}}$ $a^{a} c^{c} = {(a c)}^{b} . . . . . . . . . . . . . . . . . (i i i)$ $(i) \times (i i) \times (i i i) :$ $(a^{a} b^{b}) (b^{b} c^{c}) (a^{a} c^{c}) = {(a b)}^{c} {(b c)}^{a} {(a c)}^{b}$ ${(a^{a} b^{b} c^{c})}^{2} = a^{b + c} b^{a + c} c^{a + b}$ $a^{a} b^{b} c^{c} = a^{\frac{b + c}{2}} b^{\frac{a + c}{2}} c^{\frac{a + b}{2}}$ $N o t c o m p l e t e . . .$
	Answered by som(math1967) last updated on 16/Jul/22

	$\frac{l n (a)}{c - b} = \frac{l n (b)}{a - c} = \frac{l n (c)}{b - a} = k (l e t)$ $l n (a) = k (c - b)$ $\Rightarrow a = e^{k (c - b)} \Rightarrow a^{a} = e^{k (a c - a b)}$ $l n (b) = k (a - c) \Rightarrow b^{b} = e^{k (b a - b c)}$ $l n (c) = k (b - a) \Rightarrow c^{c} = e^{k (c b - c a)}$ $∴ a^{a} . b^{b} . c^{c}$ $= e^{k (a c - a b) + k (b a - b c) + k (c b - c a)}$ $= e^{0} = 1$
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