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

Answer 1

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.