The value of \(\begin{vmatrix}\log_{5}10&2\\2&\log_{10}5\end{vmatrix}\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → -3
Expanding the 2 × 2 determinant:
\( \displaystyle \begin{vmatrix}\log_{5}10&2\\2&\log_{10}5\end{vmatrix} = \log_{5}10 \cdot \log_{10}5 - 2 \times 2 \)
By the reciprocal property of logarithms, \(\log_{a}b = \frac{1}{\log_{b}a}\), so
\( \displaystyle \log_{5}10 \cdot \log_{10}5 = \frac{1}{\log_{10}5} \cdot \log_{10}5 = 1 \)
Hence
\(\text{Value} = 1 - 4 = -3\)