Question

Find the domain of definition of the following function:
$y=\sqrt{{\log }_{0.5}(3x-8)-{\log }_{0.5}(x^2+4)}.$

Answer 1

$y=\sqrt{{\log }_{0.5}\frac{3x-8}{x^2+4}}$

${\log }_{0.5}\frac{3x-8}{x^2+4}\ge 0$

As the base is less than $1$

$0<\frac{3x-8}{x^2+4}\le 1\Rightarrow x>\frac{8}{3}$

Also, $3x-8\le x^2-4x^2-3x+4\ge 0\Rightarrow x\in R$

Taking intersection, we get

$x\in (\frac{8}{3},\infty )$