The value of $\begin{vmatrix}\log_510&2\\2&\log_{10}5\end{vmatrix}$ is

-3

The correct answer is Option (1) → -3

Determinant = $\begin{vmatrix} \log_{5}{10} & 2 \\ 2 & \log_{10}{5} \end{vmatrix}$

Determinant $= \log_{5}{10}\cdot \log_{10}{5} - (2\cdot 2)$

Since $\log_{a}{b}=\frac{1}{\log_{b}{a}}$, we have $\log_{5}{10}=\frac{1}{\log_{10}{5}}$

So, $\log_{5}{10}\cdot \log_{10}{5} = 1$

Thus, Determinant $= 1 - 4 = -3$