Explanation

(d)
Let $\mathrm{\Delta} = \left| \begin{matrix}
1 & 5 & \pi \\
\log_{e}e & 5 & \sqrt{5} \\
\log_{10}10 & 5 & e
\end{matrix} \right|$
$\Rightarrow \ \mathrm{\Delta} = \left| \begin{matrix}
1 & 5 & \pi \\
1 & 5 & \sqrt{5} \\
1 & 5 & e
\end{matrix} \right| = 5\left| \begin{matrix}
1 & 1 & \pi \\
1 & 1 & \sqrt{5} \\
1 & 1 & e
\end{matrix} \right|$ $(\because\log_{a}{a = 1)}$
$= 0\$ $(\because$ two columns are identical)