Question

If ${\log }_{0.04}(x-1)\ge {\log }_{0.2}(x-1),$ then $x$ belongs to the interval

• A. $(1,2]$
• B. $(-\infty ,2]$
• C. $[2,\infty )$
• D. none of these

Answer 1

The correct option is C $[2,\infty )$

${\log }_{0.04}(x-1)\ge {\log }_{0.2}(x-1)$
$\Rightarrow {\log }_{(0.2{)}^2}(x-1)\ge {\log }_{0.2}(x-1)$
$\Rightarrow \frac{1}{2}{\log }_{0.2}(x-1)\ge {\log }_{0.2}(x-1)$
$\Rightarrow {\log }_{0.2}(x-1)\ge 2{\log }_{0.2}(x-1)$
$\Rightarrow {\log }_{0.2}(x-1)\le 0\Rightarrow (x-1)\ge (0.2{)}^0\Rightarrow x\ge 2\Rightarrow x\in [2,\infty )$