log⁡e log⁡e2 log⁡e3log⁡e2 log⁡e3 log⁡e4log⁡e3 log⁡e4 log⁡e5 is equal to

log⁡elog⁡e2log⁡e3log⁡e2log⁡e3log⁡e4log⁡e3log⁡e4log⁡e5=log⁡e2log⁡e3log⁡e2log⁡e3log⁡e4log⁡e3log⁡e4log⁡e5log⁡e=123234345=211211311 Applying C2→C2−C1C3→C3−C2=0(∴ Two columns are identical )

log⁡e log⁡e2 log⁡e3log⁡e2 log⁡e3 log⁡e4log⁡e3 log⁡e4 log⁡e5 is equal to

log⁡elog⁡e2log⁡e3log⁡e2log⁡e3log⁡e4log⁡e3log⁡e4log⁡e5=log⁡e2log⁡e3log⁡e2log⁡e3log⁡e4log⁡e3log⁡e4log⁡e5log⁡e=123234345=211211311 Applying C2→C2−C1C3→C3−C2=0(∴ Two columns are identical )

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$|\begin{array}{lll} \log e \log e^{2} \log e^{3} \\ \log e^{2} \log e^{3} \log e^{4} \\ \log e^{3} \log e^{4} \log e^{5} \end{array}| is equal to$

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$4 \log e$

$5 \log e$

answer is A.

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Detailed Solution

$\begin{array}{l} |\begin{array}{lll} \log e & \log e^{2} & \log e^{3} \\ \log e^{2} & \log e^{3} & \log e^{4} \\ \log e^{3} & \log e^{4} & \log e^{5} \end{array}| \\ = |\begin{array}{lll} \log e & 2 \log e & 3 \log e \\ 2 \log e & 3 \log e & 4 \log e \\ 3 \log e & 4 \log e & 5 \log e \end{array}| \\ \begin{array}{l} = |\begin{array}{lll} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{array}| \\ = |\begin{array}{lll} 2 & 1 & 1 \\ 2 & 1 & 1 \\ 3 & 1 & 1 \end{array}| Applying \\ C_{2} \to C_{2} - C_{1} C_{3} \to C_{3} - C_{2} \\ = 0 (∴ Two columns are identical) \end{array} \end{array}$