Question

Which one of the two is greater?

${\log }_{2}3$ or ${\log }_{3}11$

• A. ${\log }_{2}3$
• B. ${\log }_{3}11$
• C. Both are equal
• D. Cannot be determined

Answer 1

The correct option is B ${\log }_{3}11$

Since, $2<3<4$

Taking log with base 2, we get

$\Rightarrow 1<{\log }_{2}3<{\log }_2{2}^2$

$\Rightarrow 1<{\log }_{2}3<2$
Thus $1<{\log }_{2}3<2.$

Again since, $9<11<27$

Taking log with base 3, we get

${\log }_3{3}^2<{\log }_{3}11<{\log }_3{3}^3$

$\Rightarrow 2<{\log }_{3}11<3$

We conclude that ${\log }_{3}11>{\log }_{2}3.$