Question

Evaluate $f\left(x\right)=\sqrt{{\log }_{0.5}(3x-8)-{\log }_{0.5}(x^2+4)}$?

Answer 1

Given, $f\left(x\right)=\sqrt{{\log }_{0.5}(3x-8)-lo{g}_{0.5}(x^2+4)}$

$f\left(x\right)=\sqrt{{\log }_{0.5}\frac{(3x-8)}{(x^2+4)}}$

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

$\Rightarrow \frac{(3x-8)}{(x^2+4)}\le 1$

$\Rightarrow 3x-8\le x^2+4$

$\Rightarrow x^2-3x+12\ge 0$

$\Rightarrow xϵR$

And for, ${\log }_{0.5}3x-8$

$\Rightarrow 3x-8>0$

$\Rightarrow x>\left(\frac{8}{3}\right)$

And for, ${\log }_{0.5}{x}^2+4$

$\Rightarrow x^2+4>0$

$\Rightarrow xϵR$

Taking intersection, we get $xϵ(\frac{8}{3},\infty )$