# Solve the following equations

Question:

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

(a) 7

(b) 10

(c) 1

(d) 17

Solution:

(b) 10

$\mid \log _{3} 512 \quad \log _{4} 3$

$\log _{3} 8 \quad \log _{4} 9|\times| \log _{2} 3 \quad \log _{8} 3$

$\log _{3} 4 \quad \log _{3} 4 \mid$

$=\left|\log _{3} 2^{9} \quad \log _{2^{2}} 3\right|$

$\log _{3} 2^{3} \quad \log _{2^{2}} 3^{3}|\times| \log _{2} 3 \quad \log _{2^{3}} 3$

$\log _{3} 2^{2} \quad \log _{3} 2^{2}$

$=\mid 9 \log _{3} 2 \quad \frac{1}{2} \log _{2} 3$

$3 \log _{3} 2 \quad \frac{1}{2} \times 2 \log _{2} 3|\times| \log _{2} 3 \quad \frac{1}{3} \log _{2} 3$

$2 \log _{3} 2 \quad 2 \log _{3} 2 \mid$

$\left[\because \quad \log _{b^{m}} a^{n}=\frac{n}{m} \log _{b} a\right]$

$=\left(\left(9 \log _{3} 2 \times \log _{2} 3\right)-\left(3 \log _{3} 2 \times \frac{1}{2} \log _{2} 3\right)\right) \times\left(\left(\log _{2} 3 \times 2 \log _{3} 2\right)-\left(\frac{1}{3} \log _{2} 3 \times 2 \log _{3} 2\right)\right)$                  $\left[\because \log _{m} n \times \log _{n} m=1\right]$

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

$=\frac{15}{2} \times \frac{4}{3}=10$