Question

Solve the following inequality:
$2{\log }_{5}x-{\log }_{x}125<1$

Answer 1

$2\frac{\log x}{\log 5}-3\frac{\log 5}{\log x}<1\frac{2{\left(\log x\right)}^2-3{\left(\log 5\right)}^2}{\log x\log 5}<1Forx>12{\left(\log x\right)}^2-3{\left(\log 5\right)}^2<\log x\log 52{\left(\log x\right)}^2-\log x\log 5-3{\left(\log 5\right)}^2<02{\left(\log x\right)}^2+2\log x\log 5-3\log x\log 5-3{\left(\log 5\right)}^2<02\log x(\log x+\log 5)-3\log 5(\log x+\log 5)<0(\log x+\log 5)(2\log x-3\log 5)<0x\in (\frac{1}{5},5^{3/2})$

Hence, $x\in (1,5^{3/2})$

For $x\in (0,1)$

$2{\left(\log x\right)}^2-3{\left(\log 5\right)}^2>\log x\log 52{\left(\log x\right)}^2-\log x\log 5-3{\left(\log 5\right)}^2>0(\log x+\log 5)(2\log x-3\log 5)>0x\in (-\infty ,\frac{1}{5})\cup (5^{3/2},\infty )$

Hence $x\in (0,\frac{1}{5})$

Taking union, we get $x\in (0,\frac{1}{5})\cup (1,5^{3/2})$