Question

The domain of the function ${\log }_2\left\{{\log }_{1/2}(x^2+4x+4)\right\}$ is given by $(-3,-1)-k.$
what is k?

• A. $-2$
• B. $-2.5$
• C. $-1.5$
• D. none

Answer 1

The correct option is A $-2$

Before doing this question the following points should be noted
carefully:
${\log }_{a}x$ is defined if $x>0\text{and}x\ne 0$ and $a>0\text{and}a\ne 1$
${\log }_{a}x>{\log }_{a}y\iff x>y,ifa>1$
$\iff x<y,ifa<1$
Hence we must have$.{\log }_{1/2}(x^2+4x+4)>0={\log }_{1/2}1$
...(1)
and $x^2+4x+4\ne 0$ ...(2)
(1)$\Rightarrow x^2+4x+4<1\text{as}a<1$
$\Rightarrow x^2+4x+3<0$
$\Rightarrow (x+3)(x+1)<0$
$\therefore -3<x<-1$
$\therefore xϵ(-3,-1)$
(2)$\Rightarrow {(x+2)}^2\ne 0\therefore x\ne -2.$
Excluding -2 we can say $xϵ(-3,-2)\cup (-2,-1).$ Which is required domain.