Question

If ${\log }_{b}a+{\log }_{c}a+{\log }_{a}b+{\log }_{a}c+{\log }_{b}c+{\log }_{c}b=3$ (where $a,b,c$ are different positive real numbers $\ne 1$ ), then find the value of $abc$

Answer 1

$\frac{{\left(\log a\right)}^2}{\log b\log c}+\frac{{\left(\log b\right)}^2}{\log a\log c}+\frac{{\left(\log c\right)}^2}{\log a\log b}=3$
$\Rightarrow {\left(\log a\right)}^3+{\left(\log b\right)}^3+{\left(\log c\right)}^3-3\log a\log b\log c=0$
$\Rightarrow \log a+\log b+\log c=0(asa,b.c,aredistinct)$
$\Rightarrow \log abc=0$
$\Rightarrow abc=1$
Ans: 1