Evaluate- - 1 logxy logxz- logyx 1 logyz- logzx logzy 1- - -x-y-z being -ive-

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Properties of Determinants

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Question

Evaluate: $∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & l o g_{x} y & l o g_{x} z l o g_{y} x & 1 & l o g_{y} z l o g_{z} x & l o g_{z} y & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$
x,y,z being +ive.

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∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & {log}_{x} y & {log}_{x} z {log}_{y} x & 1 & {log}_{y} z {log}_{z} x & {log}_{z} y & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & {log}_{x} y & {log}_{x} z {log}_{y} x & 1 & {log}_{y} z {log}_{z} x & {log}_{z} y & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

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⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & {log}_{x} y & {log}_{x} z {log}_{y} x & 1 & {log}_{y} z {log}_{z} x & {log}_{z} y & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

|\begin{matrix} a + x & y & z \\ x & a + y & z \\ x & y & a + z \end{matrix}|

a^{x - y} * b^{y - z} * a^{y - x} * b^{z - y} * 2^{5}

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