Question

Let the domain of x for the given equation $lo{g}_{5x+4}\left(x^2\right)\le lo{g}_{5x+4}(2x+3)$ be $(m,n]$. Find $m+n$ ?

Answer 1

${\log }_{5x+4}\left(x^2\right)\le {\log }_{5x+4}(2x+3)$
Given inequality is equal to following inequalities
$\Rightarrow 5x+4>1,x^2>0,x^2\le 2x+3$
$\Rightarrow x>-\frac{3}{5},x>0,(x-3)(x+1)\le 0$
$\Rightarrow x>-\frac{3}{5},x>0,x\in [-1,3]$
$\Rightarrow x\in (0,3]$