Question

If ${\log }_{0.2}(x-1)>{\log }_{0.04}(x+5)$ then

• A. $-1<x<4$
• B. $2<x<3$
• C. $1<x<4$
• D. $1<x<3$

Answer 1

The correct option is C $1<x<4$

${\log }_{0.2}(x-1)>{\log }_{0.04}(x+5)$
$\Rightarrow {\log }_{0.2}(x-1)>{\log }_{(0.2{)}^2}(x+5)\Rightarrow {\log }_{0.2}(x-1)>\frac{1}{2}{\log }_{0.2}(x+5)$
$\Rightarrow (x-1{)}^2<(x+5)\Rightarrow x^2-3x-4<0\Rightarrow (x+1\left)\right(x-4)<0\Rightarrow -1<x<4$

But $x>1$ for ${\log }_{0.2}(x-1)$ to be defined.
Thus, $1<x<4$.