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### Answers (1)

Answer:

Correct option (b)

Hint:

Use logarithm properties

Solution:

$\left|\begin{array}{cc} \log _{3}^{512} & \log _{4}^{3} \\ \log _{3}^{8} & \log _{4}^{9} \end{array}\right| \times\left|\begin{array}{ll} \log _{2}^{3} & \log _{8}^{3} \\ \log _{3}^{4} & \log _{3}^{4} \end{array}\right|$

$=\left|\begin{array}{cc} \log _{3}{ }^{2^{9}} & \log _{4}^{3} \\ \log _{3}^{2^{3}} & \log _{4}^{3^{2}} \end{array}\right| \times\left|\begin{array}{cc} \log _{2}^{3} & \log _{8}^{3} \\ \log _{3}^{2^{2}} & \log _{3}^{2^{2}} \end{array}\right|$

$=\left|\begin{array}{cc} 9\log _{3}{ }^{2} & \log _{4}^{3} \\ 3\log _{3}^{2} & 2\log _{4}^{3} \end{array}\right| \times\left|\begin{array}{cc} \log _{2}^{3} & \log _{8}^{3} \\ 2\log _{3}^{2} & 2\log _{3}^{2} \end{array}\right|$

$=\left[\left(9 \times \frac{\log 2}{\log 3} \times 2 \times \frac{\log 3}{\log 4}\right)-3 \times \frac{\log 2}{\log 3} \times \frac{\log 3}{\log 4}\right] \times\left[\frac{\log 3}{\log 2} \times 2 \times \frac{\log 2}{\log 3}-\frac{\log 3}{\log 8} \times 2 \times \frac{\log 2}{\log 3}\right]$

$=\left(18 \times \frac{\log 2}{2 \log 2}-3 \times \frac{\log 2}{2 \log 2}\right)\left(2-2 \times \frac{1}{3 \log 2} \times \log 2\right)$

$=(9-\frac{3}{2})(2-\frac{2}{3})$

$=\frac{15}{2}\times \frac{4}{3}$

$=10$

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