Question

What values of x satisfy the following inequality :

${\log }_7(x+5)>{\log }_5(x+5)$ ?

Answer 1

By change of base, the inequality gives

$\frac{\log (x+5)}{\log 7}>\frac{\log (x+5)}{\log 5}$

This is true exactly when $\log (x+5)$ is negative, meaning that

$\log (x+5)<0⟹x+5<1⟹x<-4$

And, since, $x+5$ must be positive, $x>-5$

Therefore, final solution set is $-5<x<-4$.