Question

Evaluate the determinant $\Delta=\left|\begin{matrix}log_{3} \,512&log_{4} \,3\\ log_{3} \,8 &log_{4} \,9\end{matrix}\right|$

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

Answer 1

Answer: (A)

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