Question

Evaluate the determinant $\Delta =\left|\begin{array}{cc}lo{g}_{3}512& lo{g}_{4}3\\ lo{g}_{3}8& lo{g}_{4}9\end{array}\right|$

• A. $15/2$
• B. $12$
• C. $14/3$
• D. $6$

Answer 1

Answer: (A)

We have, $\Delta =\left|\begin{array}{cc}lo{g}_{3}512& lo{g}_{4}3\\ lo{g}_{3}8& lo{g}_{4}9\end{array}\right|\Rightarrow \Delta =\left|\begin{array}{cc}lo{g}_3{2}^9& lo{g}_{2^2}3\\ lo{g}_3{2}^3& lo{g}_{2^2}{3}^2\end{array}\right|$
$\Rightarrow \Delta =\left|\begin{array}{cc}9lo{g}_{3}2& \frac{1}{2}lo{g}_{2}3\\ 3lo{g}_{3}2& \frac{2}{2}lo{g}_{2}3\end{array}\right|$ $\left[\because lo{g}_{a^p}{m}^n=\frac{n}{p}lo{g}_{a}m\right]$
$\Rightarrow \Delta =\left(9lo{g}_{3}2\right)\times \left(lo{g}_{2}3\right)-\left(\frac{1}{2}lo{g}_{2}3\right)\left(3lo{g}_{3}2\right)$
$\Rightarrow \Delta =9\left(lo{g}_{3}2\times lo{g}_{2}3\right)-\frac{3}{2}\left(lo{g}_{2}3\times lo{g}_{3}2\right)$
$\Rightarrow \Delta =9-\frac{3}{2}\Rightarrow \Delta =\frac{15}{2}$ $\left[\because lo{g}_{b}a\times lo{g}_{a}b=1\right]$