Question

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

• A. $(1,2]$
• B. $(-\infty ,2)$
• C. $[2,+\infty ]$
• D. $Noneofthese$

Answer 1

The correct option is C $[2,+\infty ]$

$lo{g}_{0.04}(x-1)\ge lo{g}_{0.2}(x-1)$ ....(i)
For log to be defined $x-1>0\Rightarrow x>1$
$From\left(i\right),lo{g}_{(0.2{)}^2}(x-1)\ge lo{g}_{0.2}(x-1)$
$\Rightarrow \frac{1}{2}lo{g}_{0.2}(x-1)\ge lo{g}_{0.2}(x-1)\Rightarrow \sqrt{x-1}\le (x-1)\Rightarrow \sqrt{x-1}(1-\sqrt{x-1})\le 0\Rightarrow 1-\sqrt{x-1}\le 0\Rightarrow \sqrt{x-1}\ge 1\Rightarrow x\ge 2,\therefore xϵ[2,\infty ).$