Question

Solve the following inequality:
$3^{\log x+2}<3^{\log x^2+5}-2$

Answer 1

$3^{logx+2}<2^{log{x}^2+5}-2\Rightarrow 3^{logx}=a\Rightarrow 9a<243a^2-2\Rightarrow 243a^2-9a-2>0\Rightarrow 243a^2-27a+18a-2>0\Rightarrow 27a(9a-1)+2(9a-1)>0\Rightarrow (27a+2)(9a-1)>0$

$a\in (-\infty ,\frac{-2}{27})\cup (\frac{1}{9},\infty )$

$a$ cannot be negative.

Therefore, $3^{logx}\in (\frac{1}{9},\infty )logx\in (-2,\infty )x\in (\frac{1}{100},\infty ).$