Given-that-log-a-b-c-log-b-c-a-log-c-a-b-find-the-value-of-a-a-b-b-c-c-

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

{\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}