Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
$lo{g}_{1/2}\left(lo{g}_{3}cos\right(\pi /6)-lo{g}_{3}sin(\pi /6\left)\right)$

Answer 1

We know that $log\left(a\right)-log\left(b\right)=log\left(a/b\right)$

$\Rightarrow {\log }_3\cos \left(\frac{\pi }{6}\right)-{\log }_3\sin \left(\frac{\pi }{6}\right)={\log }_3\cot \frac{\pi }{6}={\log }_3\sqrt{3}=\frac{1}{2}$

$\therefore$ ${\log }_{\frac{1}{2}}\frac{1}{2}=1$