lg^2(x)+2lg(10)(x)>3 inequation

Given the inequality:
$$x 2 \log{\left(10 \right)} + \log{\left(x \right)}^{2} > 3$$
To solve this inequality, we must first solve the corresponding equation:
$$x 2 \log{\left(10 \right)} + \log{\left(x \right)}^{2} = 3$$
Solve:
$$x_{1} = 0.59161255434212$$
$$x_{2} = 0.262102533701829$$
$$x_{1} = 0.59161255434212$$
$$x_{2} = 0.262102533701829$$
This roots
$$x_{2} = 0.262102533701829$$
$$x_{1} = 0.59161255434212$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.262102533701829$$
=
$$0.162102533701829$$
substitute to the expression
$$x 2 \log{\left(10 \right)} + \log{\left(x \right)}^{2} > 3$$
$$0.162102533701829 \cdot 2 \log{\left(10 \right)} + \log{\left(0.162102533701829 \right)}^{2} > 3$$

3.31067566481757 + 0.324205067403657*log(10) > 3

one of the solutions of our inequality is:
$$x < 0.262102533701829$$

 _____           _____          
      \         /
-------ο-------ο-------
       x2      x1

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 0.262102533701829$$
$$x > 0.59161255434212$$