Question

Solve the following inequality:
${\log }_{0.5}^{2}x+6⩾5{\log }_{0.5}x$

Answer 1

$\frac{{\log }^{2}x}{{\log }^{2}2}+6\ge \frac{-5\log x}{\log 2}{\log }^{2}x+6{\log }^{2}2\ge -5\log x\log 2{\log }^{2}x+5\log x\log 2x+6{\log }^{2}2\ge 0(\log x+2\log 2)(\log x+3\log 2)\ge 0x\in (-\infty ,\frac{1}{8}]\cup [\frac{1}{4},\infty )$

Also for $\log xx>0$

Taking intersection, we get $x\in [\frac{1}{4},\infty )$