CAT 2020 Slot 2 QA Question & Solution
AlgebraMedium
Question
The value of $\log_{a}({\frac{a}{b}})+\log_{b}({\frac{b}{a}})$, for $1
Options
0
-1
1
-0.5
Solution
On expanding the expression we get $1-\log_ab+1-\log_ba$
$or\ 2-\left(\log_ab+\frac{1}{\log_ab}\right)$
Now applying the property of AM>=GM, we get that $\frac{\left(\log_ab+\frac{1}{\log_ab}\right)}{2}\ge1\ or\ \left(\log_ab+\frac{1}{\log_ab}\right)\ge2$ Hence from here we can conclude that the expression will always be equal to 0 or less than 0. Hence any positive value is not possible. So 1 is not possible.