Question

Find the domain of the following :

${\log }_{\frac{x+6}{3}}\left[{\log }_2\left(\frac{x-1}{2+x}\right)\right]>0$

Answer 1

Case I: $\frac{x+6}{3}>1\Rightarrow x>-3$ ...(i)

Case II: $\frac{x-1}{x+2}>0\Rightarrow xϵ(-\infty ,-2)\cup (1,\infty )$ ...(ii)

Case III: ${\log }_2\frac{(x-1)}{(2+x)}>0$

$\Rightarrow \left(\frac{x-1}{2+x}\right)>1\Rightarrow \frac{x-1-2-x}{2+x}>0$

$\Rightarrow \frac{-3}{x+2}>0\Rightarrow \frac{1}{x+2}<0\Rightarrow x<-2$ ...(iii)

$\therefore$ From above 3 cases, we get that

$-3<x<-2$