Question

The value of $\frac{{\log }_{5}9\cdot {\log }_{7}5\cdot {\log }_{3}7}{{\log }_3\sqrt{6}}+\frac{1}{{\log }_4\sqrt{6}}$ is co-prime with

• A. 3
• B. 4
• C. 5
• D. 9

Answer 1

Answer: (A), (C), (D)

$\frac{{\log }_{5}9\cdot {\log }_{7}5\cdot {\log }_{3}7}{{\log }_3\sqrt{6}}+\frac{1}{{\log }_4\sqrt{6}}$
$\frac{{\log }_{3}9}{\frac{1}{2}{\log }_{3}6}+\frac{1}{\frac{1}{2}{\log }_{4}6}$
$\frac{2\cdot 2}{{\log }_{3}6}+\frac{2}{{\log }_{4}6}$
$4\cdot {\log }_{6}3+2{\log }_{6}4\Rightarrow {\log }_6{3}^4+{\log }_6{4}^2$
$={\log }_6\left(3^4\cdot 2^4\right)=4$