Question

If $\frac{\log a}{b-c}=\frac{\log b}{c-a}=\frac{\log c}{a-b}$, then find $a^a\times b^b\times c^c$

Answer 1

To find $a^a{\times b}^b\times c^c$

Let $\frac{\log a}{b-c}=\frac{\log b}{c-a}=\frac{\log c}{a-b}=k$

$\log a=k(b-c)\Rightarrow a={10}^{k(b-c)}$

$\log b=k(c-a)\Rightarrow b={10}^{k(c-a)}$

$\log c=k(a-b)\Rightarrow c={10}^{k(a-b)}$

$\Rightarrow a^a={10}^{ak(b-c)}$

$\Rightarrow b^b={10}^{bk(c-a)}$

$\Rightarrow c^c={10}^{ck(a-b)}$

$\Rightarrow a^a{b}^b{c}^c={10}^{k\left[a\right(b-c)+b(c-a)+c(a-b\left)\right]}$

$\Rightarrow a^a{b}^b{c}^c={10}^{k\times 0}=1$