Question

Solve:

$\frac{{\log }_{xy}xy}{{\log }_{xy}xyz}+\frac{{\log }_{yz}yz}{{\log }_{yz}xyz}+\frac{{\log }_{zx}zx}{{\log }_{zx}xyz}$

Answer 1

We have,

$\Rightarrow \frac{{\log }_{xy}xy}{{\log }_{xy}xyz}+\frac{{\log }_{yz}yz}{{\log }_{yz}xyz}+\frac{{\log }_{zx}zx}{{\log }_{zx}xyz}$

We know that

$\frac{{\log }_{a}m}{{\log }_{a}n}={\log }_{n}m$

Therefore,

$\Rightarrow {\log }_{xyz}xy+{\log }_{xyz}yz+{\log }_{xyz}zx$

$\Rightarrow {\log }_{xyz}\left(x^2{y}^2{z}^2\right)$

$\Rightarrow {\log }_{xyz}(xyz{)}^2$

$\Rightarrow 2{\log }_{xyz}\left(xyz\right)$

$\Rightarrow 2$

Hence, this is the answer.