Question

If ${\log }_{0.2}(x-1)>{\log }_{0.04}(x+5)$, then which of the following is true?

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

Answer 1

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

We have,

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

$\Rightarrow {\log }_{0.2}(x-1)>{\log }_{(0.2{)}^2}(x+5)$

$\Rightarrow {\log }_{0.2}(x-1)>\frac{1}{2}{\log }_{0.2}(x+5)$

$\Rightarrow 2{\log }_{0.2}(x-1)>{\log }_{0.2}(x+5)$

$\Rightarrow {\log }_{0.2}(x-1{)}^2>{\log }_{0.2}(x+5)$

$\Rightarrow (x-1{)}^2<x+5$ $[\because {\log }_{a}x>{\log }_{a}y\Rightarrow x<y$, if $0<a<1]$

$\Rightarrow x^2-2x+1<x+5$
$\Rightarrow x^2-3x-4<0$
$\Rightarrow x^2-4x+x-4<0$
$\Rightarrow x(x-4)+1(x-4)<0$
$\Rightarrow (x-4)(x+1)<0$
$\Rightarrow x\in (-1,4)$

But $x>1$
$\Rightarrow x\in (1,4)$