Question

What values of x satisfy the following inequality :

${\log }_2(x+1)>{\log }_4\left(x^2\right)$ ?

Answer 1

Here, the bases are different, but they are related by the fact that $4=2^2$.

Rewriting the inequality to use 4 as a base gives

${\log }_4\left(\right(x+1{)}^2)>{\log }_4(x^2)$

So, $(x+1{)}^2>x^2$, implying that $2x+1>0⟹x>-\frac{1}{2}$.

Additionally, the arguments of each logarithm must be positive, which excludes the case $x=0$.

Therefore, the final solution set is $x>-\frac{1}{2},x\ne 0$.