The value of the determinant
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z l o g 2 x & l o g 2 y & l o g 2 z l o g 3 x & l o g 3 y & l o g 3 z \end{matrix} ∣ ∣ ∣ ∣$ is

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log (xyz)

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log (6 xyz)

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6 log (xyz)

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Solution

The correct option is A
0

$Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z l o g 2 x & l o g 2 y & l o g 2 z l o g 3 x & l o g 3 y & l o g 3 z \end{matrix} ∣ ∣ ∣ ∣$

$Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z l o g 2 + l o g x & l o g 2 + l o g y & l o g 2 + l o g z l o g 3 + l o g x & l o g 3 + l o g y & l o g 3 + l o g z \end{matrix} ∣ ∣ ∣ ∣$
(Using log (ab) =log a + log b )
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z l o g 2 & l o g 2 & l o g 2 l o g 3 & l o g 3 & l o g 3 \end{matrix} ∣ ∣ ∣ ∣$
$(R_{3} \to R_{3} - R_{1}, R_{2} \to R_{2} - R_{1})$
$= (l o g 2) (l o g 3) ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z 1 & 1 & 1 1 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0$

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The value of the determinant
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z l o g 2 x & l o g 2 y & l o g 2 z l o g 3 x & l o g 3 y & l o g 3 z \end{matrix} ∣ ∣ ∣ ∣$ is

Q. The value of the determinant

Δ = ∣ ∣ ∣ ∣ \begin{matrix} log x log 2 x log 3 x \end{matrix} \begin{matrix} log y log 2 y log 3 y \end{matrix} \begin{matrix} log z log 2 z log 3 z \end{matrix} ∣ ∣ ∣ ∣

Q. For positive numbers x, y and z the numerical value of the determinant

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & l o g_{x} y & l o g_{x} z l o g_{y} x & 1 & l o g_{y} z l o g_{z} x & l o g_{z} y & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

For positive numbers x, y and z the numerical value of the following determinant

$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & {l o g}_{x} y & {l o g}_{x} z {l o g}_{y} x & 1 & {l o g}_{y} z {l o g}_{z} x & {l o g}_{z} y & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ is

Q. If

x

y

z

are

\neq 0, \neq 1

the value of

∣ ∣ ∣ ∣ \begin{matrix} log x & log y & log z log 2 x & log 2 y & log 2 z log 3 x & log 3 y & log 3 z \end{matrix} ∣ ∣ ∣ ∣

is equal to

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The value of the determinant Δ=∣∣ ∣∣log xlog ylog zlog 2xlog 2ylog 2zlog 3xlog 3ylog 3z∣∣ ∣∣ is

The value of the determinant
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z l o g 2 x & l o g 2 y & l o g 2 z l o g 3 x & l o g 3 y & l o g 3 z \end{matrix} ∣ ∣ ∣ ∣$ is