The value of the determinantloga x-y logay-z loga z-x logby-z logb z-x logb x-y logc z-x logc x-y logc y-z is

Log_n m =ln m/ln n Δ = log_a (x/y ) amp; log_a(y/z ) amp; log_a (z/x ) log_b(y/z ) amp;log_b (z/x ) amp;log_b (x/y ) log_c (z/x ) amp; log ...

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Byju's Answer

Standard XIII

Mathematics

Evaluation of a Determinant

The value of ...

Question

The value of the determinant
$∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} l o g_{a} (\frac{x}{y}) & l o g_{a} (\frac{y}{z}) & l o g_{a} (\frac{z}{x}) l o g_{b} (\frac{y}{z}) & l o g_{b} (\frac{z}{x}) & l o g_{b} (\frac{x}{y}) l o g_{c} (\frac{z}{x}) & l o g_{c} (\frac{x}{y}) & l o g_{c} (\frac{y}{z}) \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣$ is

$l o g_{a} (x y z) + l o g_{b} (x y z) + l o g_{c} (x y z)$

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$a l o g_{e} x + b l o g_{e} y + c l o g_{e} z$

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-1

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None of these

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Solution

The correct option is D
None of these

$l o g_{n} m = \frac{l n m}{l n n} Δ = ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} l o g_{a} (\frac{x}{y}) & l o g_{a} (\frac{y}{z}) & l o g_{a} (\frac{z}{x}) l o g_{b} (\frac{y}{z}) & l o g_{b} (\frac{z}{x}) & l o g_{b} (\frac{x}{y}) l o g_{c} (\frac{z}{x}) & l o g_{c} (\frac{x}{y}) & l o g_{c} (\frac{y}{z}) \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣$
$Δ = \frac{1}{l n a . l n b . l n c} ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} l n (\frac{x}{y}) & l n (\frac{y}{z}) & l n (\frac{z}{x}) l n (\frac{y}{z}) & l n (\frac{z}{x}) & l n (\frac{x}{y}) l n (\frac{z}{x}) & l n (\frac{x}{y}) & l n (\frac{y}{z}) \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣$
$l n \frac{m}{n} = l n m - l n n$
$Δ = \frac{1}{l n a . l n b . l n c} ∣ ∣ ∣ ∣ ∣ \begin{matrix} (l n x - l n y) & (l n y - l n z) & (l n z - l n x) (l n y - l n z) & (l n z - l n x) & (l n x - l n y) (l n z - l n x) & (l n x - l n y) & (l n y - l n z) \end{matrix} ∣ ∣ ∣ ∣ ∣$
$Δ = \frac{1}{l n a . l n b . l n c} ∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & 0 & 0 (l n y - l n z) & (l n z - l n x) & (l n x - l n y) (l n z - l n x) & (l n x - l n y) & (l n y - l n z) \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0 (R_{1} \to R_{1} + R_{2} + R_{3})$

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∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} l o g_{a} (\frac{x}{y}) & l o g_{a} (\frac{y}{z}) & l o g_{a} (\frac{z}{x}) l o g_{b} (\frac{y}{z}) & l o g_{b} (\frac{z}{x}) & l o g_{b} (\frac{x}{y}) l o g_{c} (\frac{z}{x}) & l o g_{c} (\frac{x}{y}) & l o g_{c} (\frac{y}{z}) \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣

Q. The value of the determinant

∣ ∣ ∣ ∣ ∣ \begin{matrix} l o g_{a} (x / y) & l o g_{a} (y / z) & l o g_{a} (z / x) l o g_{b} (y / z) & l o g_{b} (z / x) & l o g_{b} (x / y) l o g_{c} (z / x) & l o g_{c} (x / y) & l o g_{c} (y / z) \end{matrix} ∣ ∣ ∣ ∣ ∣

Q. the value of the Determinant

∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} {log}_{a} (\frac{x}{y}) & {log}_{a} (\frac{y}{z}) & {log}_{a} (\frac{z}{x}) {log}_{b} (\frac{y}{z}) & {log}_{b} (\frac{z}{x}) & {log}_{b} (\frac{x}{y}) {log}_{c} (\frac{z}{x}) & {log}_{c} (\frac{x}{y}) & {log}_{c} (\frac{y}{z}) \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣

is equal to

Q. The minimum value of

| z - 1 | + | z |

for complex values of z is

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The value of the determinant ∣∣ ∣ ∣ ∣ ∣∣loga(xy)loga(yz)loga(zx)logb(yz)logb(zx)logb(xy)logc(zx)logc(xy)logc(yz)∣∣ ∣ ∣ ∣ ∣∣ is