Question

The solution set of the inequality $({\log }_{10}x{)}^4-13({\log }_{10}x{)}^2+36>0$ is

• A. $(1000,\infty )$
• B. $(\frac{1}{100},100)\cup (1000,\infty )$
• C. $(0,100)\cup (1000,\infty )$
• D. $(0,\frac{1}{1000})\cup (\frac{1}{100},100)\cup (1000,\infty )$

Answer 1

The correct option is D $(0,\frac{1}{1000})\cup (\frac{1}{100},100)\cup (1000,\infty )$

Given,
$({\log }_{10}x{)}^4-13({\log }_{10}x{)}^2+36>0,x>0$
$\Rightarrow \left[\right({\log }_{10}x{)}^2-4\left]\right[({\log }_{10}x{)}^2-9]>0$
$\Rightarrow ({\log }_{10}x+2)({\log }_{10}x-2)({\log }_{10}x+3)({\log }_{10}x-3)>0$
$\Rightarrow {\log }_{10}x\in (-\infty ,-3)\cup (-2,2)\cup (3,\infty )$
$\Rightarrow x\in (0,\frac{1}{1000})\cup (\frac{1}{100},100)\cup (1000,\infty )$