Question

$\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|$=

• A. $\log \left(xyz\right)$
• B. $\log (xy+yz+zx)$
• C. $0$
• D. $\log (x+y+z)$

Answer 1

The correct option is C $0$

$\begin{array}{c}\left|\begin{array}{c}1{\log }_x^y{\log }_x^z\\ {\log }_y^{x}1{\log }_y^z\\ {\log }_z^x{\log }_z^{y}1\end{array}\right|\\ \Rightarrow 1\left[1-\frac{{\log }_y^z}{{\log }_y^z}\right]-{\log }_x^y\left[{\log }_y^x-\frac{{\log }_z^x}{{\log }_z^y}\right]-{\log }_x^z\\ \left[\frac{{\log }_y^x}{{\log }_y^z}-{\log }_z^x\right]\\ \Rightarrow 0-{\log }_x^y\left[{\log }_y^x-{\log }_2^{\left(x-y\right)}\right]-{\log }_x^2\left[{\log }_y^{\left(x-z\right)}-{\log }_2^x\right]\\ -1+1=0Ans.\end{array}$