Question
Using properties of determinant, show that :
$\left|\begin{array}{ccc}1 & \log _x y & \log _x z \\ \log _y x & 1 & \log _y z \\ \log _2 x & \log _x y & 1\end{array}\right|=0$
$\left|\begin{array}{ccc}1 & \log _x y & \log _x z \\ \log _y x & 1 & \log _y z \\ \log _2 x & \log _x y & 1\end{array}\right|=0$
$\cdots\left[\because \log _e b=\frac{\log _e b}{\log _e c}\right]$
Taking $\frac{1}{\log _e x}, \frac{1}{\log _e y}, \frac{1}{\log _e \mathrm{z}}$ common from $\mathrm{R}_1$,
$R_2, R_3$ respectively, we get
L.H.S.
$=\frac{1}{\log _e x \cdot \log _e y \cdot \log _e z}\left|\begin{array}{lll}\log _e x & \log _e y & \log _e z \\ \log _e x & \log _e y & \log _e z \\ \log _e x & \log _e y & \log _e z\end{array}\right|$
$=\frac{1}{\log _{\mathrm{e}} x \cdot \log _{\mathrm{e}} y_{\cdot} \log _{\mathrm{e}} z}(0)$
$\ldots\left[\because \mathbf{R}_1, \mathbf{R}_2, \mathbf{R}_3\right.$ are identical $]$
$=0=$ R.H.S.
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