Question

If $x\in R$ satisfies $(lo{g}_{10}100x{)}^2+(lo{g}_{10}10x{)}^2+lo{g}_{10}x\le 14$ then $x$ contains the interval

• A. (0,10)
• B. $[{10}^{\frac{-9}{2}},10]$
  
• C. $(0,\infty )$
  
• D. $(-1,\infty )$

Answer 1

The correct option is B

$[{10}^{\frac{-9}{2}},10]$

Writing $u=lo{g}_{10}x$, the given inequality can be written as
$(2+u{)}^2+(1+u{)}^2+u\le 14\Rightarrow 2u^2+7u-9\le 0\Rightarrow (2u+9)(u-1)\le 0\Rightarrow \frac{-9}{2}\le u\le 1\Rightarrow \frac{-9}{2}\le lo{g}_{10}x\le 1\Rightarrow {10}^{\frac{-9}{2}}\le x\le 10$