If a- b and c are non-zero and different from 1- then the value of - loga1 logab logac- loga1-b logb1 loga1-c- loga1-c logac logc1 - is

Explanation for the correct option:Find the value of given determinant: △ = |[ log_a1 log_ab log_ac; log_a(1/b) log_b1 log_a(1/c); log_a(1/ ...

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Byju's Answer

Standard XII

Mathematics

Definition of a Determinant

If a, b and c...

Question

If $a, b$ and $c$ are non-zero and different from $1$ , then the value of $|\begin{matrix} \log_{a} 1 & \log_{a} b & \log_{a} c \\ \log_{a} (\frac{1}{b}) & \log_{b} 1 & \log_{a} (\frac{1}{c}) \\ \log_{a} (\frac{1}{c}) & \log_{a} c & \log_{c} 1 \end{matrix}|$ is

$0$

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$1 + \log_{a} (a + b + c)$

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$\log_{a} (a + b + c)$

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$1$

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Solution

The correct option is A
$0$

Explanation for the correct option:
Find the value of given determinant:
$△ = |\begin{matrix} \log_{a} 1 & \log_{a} b & \log_{a} c \\ \log_{a} (\frac{1}{b}) & \log_{b} 1 & \log_{a} (\frac{1}{c}) \\ \log_{a} (\frac{1}{c}) & \log_{a} c & \log_{c} 1 \end{matrix}|$
$\Rightarrow$ $△ = |\begin{matrix} 0 & \log_{a} b & \log_{a} c \\ - \log_{a} b & 0 & - \log_{a} c \\ - \log_{a} c & \log_{a} c & 0 \end{matrix}|$
$\Rightarrow$ $△ = 0 - l o g_{a} b [0 - {(\log_{a} c)}^{2}] + \log_{a} c [- \log_{a} b \log_{a} c - 0]$
$\Rightarrow$ $△ = \log_{a} b {(\log_{a} c)}^{2} - \log_{a} b {(\log_{a} c)}^{2}$
$∴ △ = 0$
Hence, option ‘A’ is Correct.

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If a,b and c are non-zero and different from 1, then the value of loga1logablogacloga1blogb1loga1cloga1clogaclogc1 is

If $a, b$ and $c$ are non-zero and different from $1$ , then the value of $|\begin{matrix} \log_{a} 1 & \log_{a} b & \log_{a} c \\ \log_{a} (\frac{1}{b}) & \log_{b} 1 & \log_{a} (\frac{1}{c}) \\ \log_{a} (\frac{1}{c}) & \log_{a} c & \log_{c} 1 \end{matrix}|$ is