Question

The solution set of inequality $x^{({\log }_{10}x{)}^2-3({\log }_{10}x)+1}>1000$ is

• A. $(1000,\infty )$
• B. $(100,\infty )$
• C. $(0,1000)$
• D. $(-1000,\infty )$

Answer 1

The correct option is A $(1000,\infty )$

$x^{({\log }_{10}x{)}^2-3({\log }_{10}x)+1}>1000$
$\log$ is defined when $x>0$
Taking $\log$ with base $10$ both sides
$\Rightarrow [({\log }_{10}x{)}^2-3({\log }_{10}x)+1]{\log }_{10}x>3$
$\Rightarrow ({\log }_{10}x{)}^3-3({\log }_{10}x{)}^2+\left({\log }_{10}x\right)-3>0$
$\Rightarrow \left[\right({\log }_{10}x{)}^2+1\left]\right({\log }_{10}x-3)>0$
Since, $1+({\log }_{10}x{)}^2$ is always positive
$\Rightarrow {\log }_{10}x-3>0$
$\Rightarrow x>1000$
$\therefore x\in (1000,\infty )$