question_answer

Evaluate       the       determinant \[\Delta =\left| \begin{matrix}    {{\log }_{3}}512 & {{\log }_{4}}3  \\    {{\log }_{3}}8 & {{\log }_{4}}9  \\ \end{matrix} \right|.\]

Answer:

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