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 $[2,+\infty )$

${\log }_{0.04}(x-1)\ge {\log }_{0.2}(x-1)$, where $x>1$
$\Rightarrow \frac{\log (x-1)}{\log 0.04}\ge \frac{\log (x-1)}{\log 0.2}\Rightarrow \frac{\log (x-1)}{-\log 25}\ge \frac{\log (x-1)}{-\log 5}\Rightarrow \log (x-1)\le 2\log (x-1)\Rightarrow x-1\le {(x-1)}^2\Rightarrow {(x-1)}^2(x-2)\le 0$
Therefore, $x\ge 2$
Ans: C