Question

Number of integers satisfying inequality,
$\sqrt{{\log }_{3}x-1}+\frac{\frac{1}{2}{\log }_3{x}^3}{{\log }_3(\frac{1}{3})}+2>0$ is

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is B $6$

Let ${\log }_{3}x=t^2+1$
$\sqrt{t^2+1-1}+\frac{1}{2}\frac{3(t^2+1)}{-1}+2>0$
$\Rightarrow 2|t|-3(t^2+1)+4>0$
$\Rightarrow 3|t{|}^2-3|t|+|t|-1<0$
$\Rightarrow |t|\in (-\frac{1}{3},1)\Rightarrow t^2\in [0,1)$
$\Rightarrow x\in [3^1,3^2)\Rightarrow x\in [3,9)\Rightarrow 6\text{integers}$