Question

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

Answer 1

Given, ${\log }_{b}a\cdot {\log }_{c}a+{\log }_{a}b\cdot {\log }_{c}b+{\log }_{a}c\cdot {\log }_{b}c=3$

$⟹\frac{{\left(\log a\right)}^2}{\log b\log c}+\frac{{\left(\log b\right)}^2}{\log a\log c}+\frac{{\left(\log c\right)}^2}{\log a\log b}=3[\because {\log }_{a}b=\frac{\log b}{\log a}]$

$\Rightarrow {\left(\log a\right)}^3+{\left(\log b\right)}^3+{\left(\log c\right)}^3-3\log a\log b\log c=0$

$\Rightarrow \log a+\log b+\log c=0(Asa,b.c,aredistinct)$

$\Rightarrow \log abc=0,[\because \log x+\log y+\log z=\log (xyz\left)\right]$

$\Rightarrow abc=1$.