how to solve following up pgt math previous year question solution

Question : Solve following up pgt math previous year question solutionSolution :Let\[ A=\left|\begin{matrix}1&\log_x{y}&\log_x{z}\\\log_y{x}&1&\log_y{z}\\\log_z{x}&\log_z{y}&1\\\end{matrix}\right|\] \[A=1\left(1-\log_z{y}\log_y{z}\right)-\log_x{y}\left(\log_y{x}-\log_z{x}\log_y{z}\right)+\log_x{z}\left(\log_y{x}\log_z{y}-\log_z{x}\right)\] \[=1-\log_z{y}\log_y{z}-\log_x{y}\log_y{x}+\log_x{y}\log_z{x}\log_y{z}+\log_x{z}\log_y{x}\log_z{y}-\log_x{z}\log_z{x}\] By the property of logarithm \[\log_a{b}=\frac{\log_e{b}}{\log_e{a}}\] Therefore A=1-1-1+1+1-1 => A=0 Hence correct option is (1)

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