Question

For positive numbers $x,y,z$ the numerical value of
$\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|$ is :

• A. 0
• B. $\log x\log y\log z$
• C. 1
• D. 8

Answer 1

Answer: (A)

0

$\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|$
$=(1-{\log }_{z}y{\log }_{y}z)-{\log }_{x}y({\log }_{y}x-{\log }_{z}x{\log }_{y}z)+{\log }_{x}z({\log }_{y}x{\log }_{z}y-{\log }_{z}x)$
$=(1-1)-(1-{\log }_{x}y{\log }_{y}x)+({\log }_{x}z{\log }_{z}x-1)=0$
$(\therefore {\log }_{x}y\cdot {\log }_{y}x=1)$