Question

If ${\log }_2\left(\frac{x-1}{x-2}\right)>0$, then

• A. $x<2$
• B. $x>2$
• C. $x<1$
• D. $x>1$

Answer 1

The correct option is B $x>2$

For the inequality to be defined,
$\frac{x-1}{x-2}>0$
$\Rightarrow x\in (-\infty ,1)\cup (2,\infty )\cdots \left(1\right)$

Now, ${\log }_2\left(\frac{x-1}{x-2}\right)>0$
$\Rightarrow \frac{x-1}{x-2}>2^0\Rightarrow \frac{x-1}{x-2}-1>0\Rightarrow \frac{x-1-x+2}{x-2}>0\Rightarrow \frac{1}{x-2}>0\Rightarrow x>2\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$,
$x>2$