for positive numbers x,y,z the value of the determinant $$1logxylogxzlogyx1logyzlogzxlogzy1=$$

We $$have1logylogxlogzlogxlogxlogy$$ $$11ogzlogylogxlogzlogylogz$$ $$1$$ $$=1logxlogylogzlogxlogylogzlogxlogylogzlogxlogylogz=0$$

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Question

for positive numbers $x, y, z$ the value of the determinant $|\begin{matrix} 1 & \log_{x} y & \log_{x} z \\ \log_{y} x & 1 & \log_{y} z \\ \log_{z} x & \log_{z} y & 1 \end{matrix}| =$

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Solution

We have
$\begin{array}{l} |\begin{array}{l} 1 & \frac{\log y}{\log x} & \frac{\log z}{\log x} \\ \frac{\log x}{\log y} & 1 & \frac{1 og z}{\log y} \\ \frac{\log x}{\log z} & \frac{\log y}{\log z} & 1 \end{array}| \end{array} = \frac{1}{\log x \log y \log z} |\begin{array}{l} \log x & \log y & \log z \\ \log x & \log y & \log z \\ \log x & \log y & \log z \end{array}| = 0$

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Introduction to Determinants

Remember concepts with our Masterclasses.

for positive numbers x,y,z the value of the determinant 1logxylogxzlogyx1logyzlogzxlogzy1=

We have1logylogxlogzlogxlogxlogy 11ogzlogylogxlogzlogylogz 1 =1logxlogylogzlogxlogylogzlogxlogylogzlogxlogylogz=0

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for positive numbers $x, y, z$ the value of the determinant $|\begin{matrix} 1 & \log_{x} y & \log_{x} z \\ \log_{y} x & 1 & \log_{y} z \\ \log_{z} x & \log_{z} y & 1 \end{matrix}| =$