Question

Solve the following inequalities and find the range of $\log x$
(i) $lo{g}^{2}x\ge logx+2$

Answer 1

${\log }^{2}x\ge \log x+2$

$\Rightarrow {\log }^{2}x-\log x-2\ge 0$

Let $\log x=\alpha$

$\Rightarrow {\alpha }^2-\alpha -2\ge 0$

$\Rightarrow {\alpha }^2-2\alpha +\alpha -2\ge 0$

$\Rightarrow \alpha (\alpha -2)+1(\alpha -2)\ge 0$

$\Rightarrow (\alpha -2)(\alpha +1)\ge 0$

$\therefore \alpha \in (-\infty ,-1]\cup [2,\infty )$

$\therefore \log x\in (-\infty ,-1]\cup [2,\infty )$