Question

What values of x satisfy the following inequality :

${\log }_2\left({\log }_3\right(4x+1\left)\right)>{\log }_2\left({\log }_3\right(2x+3\left)\right)$ ?

Answer 1

Since the base is 2, which is greater than 1, the given inequality implies ${\log }_3(4x+1)>{\log }_3(2x+3)$.

This further implies that $4x+1>2x+3$, or $2x>2⟹x>1$.

Therefore, x must be greater than 1.

In addition, the arguments of all the logarithms must be positive, but this is indeed the case when x is greater than 1.

Therefore, the solution set is $x>1$.