Question

Find the value of $x$ which satisfies the equation, ${\log }_2\frac{x-1}{x-2}>0$

• A. $x>2$
• B. $x<2$
• C. $x>=2$
• D. none of these

Answer 1

The correct option is A $x>2$

Domain is $\left(\frac{x-1}{x-2}\right)>0\Rightarrow x\in (-\infty ,1)\cup (2,\infty )$
Given ${\log }_2\frac{x-1}{x-2}>0$
$\Rightarrow \frac{x-1}{x-2}>2^0$
$\Rightarrow \frac{x-1}{x-2}>1$
$\Rightarrow \frac{x-1}{x-2}-1>0$
$\Rightarrow \frac{x-1-x+2}{x-2}>0$

$\Rightarrow \frac{1}{x-2}>0$
$\Rightarrow x>2$