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Question
Find the value of `|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` if x, y, z ≠ 1
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Solution
Let Δ = `|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` where x, y, z ≠ 1
`log_x y = log_"e" y log_x "e"`
= `log_"e" y * 1/log_"e" x`
`log_x y = log y/logx`
Δ = `|(1, logy/logx, logz/logx),(logx/logy, 1, logz/logy),(logx/logz, logy/logz, 1)|`
= `logx/logx * logy/logy * logz/logz |(1, logy/logx, logz/logx),(logx/logy, 1, logz/logy),(logx/logz, logy/logz, 1)|`
= `1/(log x log y log z) |(log x, log y, log z),(log x, log y, log z),(log x, log y, log z)|`
= `1/(log x log y log z) xx 0`
= 0
Property 4: If two rows (columns) of a determinant are identical then its determinant value is zero.
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