A) \[0\]
B) \[1\]
C) \[xyz\]
D) \[\log \,xyz\]
Correct Answer: A
Solution :
\[\left| \begin{matrix} 1 & {{\log }_{x}}y & {{\log }_{x}}z \\ {{\log }_{y}}x & 1 & {{\log }_{y}}z \\ {{\log }_{e}}x & {{\log }_{e}}x & 1 \\ \end{matrix} \right|\] \[=1(1-{{\log }_{y}}z{{\log }_{z}}y)\] \[-{{\log }_{x}}y({{\log }_{y}}x-{{\log }_{z}}\,x{{\log }_{y}}z)\] \[+{{\log }_{x}}z({{\log }_{z}}y({{\log }_{y}}x-{{\log }_{z}}x)\] \[=(1-{{\log }_{y}}y)-{{\log }_{x}}y({{\log }_{y}}x-{{\log }_{y}}x)\] \[+{{\log }_{x}}z({{\log }_{z}}x-{{\log }_{z}}x)\] \[=(1-1)-0+0=0\]
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