Question

Which one of the two is greater ?

${\log }_{1/3}\frac{1}{2}$ or ${\log }_{1/2}\frac{1}{3}$

• A. ${\log }_{1/3}\frac{1}{2}$
• B. ${\log }_{1/2}\frac{1}{3}$
• C. Both are equal
• D. Cannot be determined

Answer 1

The correct option is B ${\log }_{1/2}\frac{1}{3}$

Let $x={\log }_{1/3}\frac{1}{2}$

Changing base of log to 3, we get

$x=\frac{{\log }_3\frac{1}{2}}{{\log }_3\frac{1}{3}}$

$\Rightarrow x=\frac{-1}{-1}{\log }_{3}2$

$\Rightarrow x={\log }_{3}2$

Since, $2<3$

${\log }_{3}2<{\log }_{3}3=1$

$\therefore x<1$

$y={\log }_{1/2}\frac{1}{3}$

Changing base of log to 2, we get

$y=\frac{{\log }_2\frac{1}{3}}{{\log }_3\frac{1}{2}}$

$y=\frac{-1}{-1}{\log }_{2}3>{\log }_{2}2=1$
$\therefore y>1$
Hence $y>x$

$\Rightarrow {\log }_{1/2}\frac{1}{3}>{\log }_{1/3}\frac{1}{2}.$