logx2+3log2x2-6log4x2<0 inequation

Given the inequality:
$$\left(2 \log{\left(x \right)} + 2 \cdot 3 \log{\left(2 x \right)}\right) - 2 \cdot 6 \log{\left(4 x \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(2 \log{\left(x \right)} + 2 \cdot 3 \log{\left(2 x \right)}\right) - 2 \cdot 6 \log{\left(4 x \right)} = 0$$
Solve:
$$x_{1} = \frac{\sqrt{2}}{32}$$
$$x_{1} = \frac{\sqrt{2}}{32}$$
This roots
$$x_{1} = \frac{\sqrt{2}}{32}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\sqrt{2}}{32}$$
=
$$- \frac{1}{10} + \frac{\sqrt{2}}{32}$$
substitute to the expression
$$\left(2 \log{\left(x \right)} + 2 \cdot 3 \log{\left(2 x \right)}\right) - 2 \cdot 6 \log{\left(4 x \right)} < 0$$
$$- 2 \cdot 6 \log{\left(4 \left(- \frac{1}{10} + \frac{\sqrt{2}}{32}\right) \right)} + \left(2 \log{\left(- \frac{1}{10} + \frac{\sqrt{2}}{32} \right)} + 2 \cdot 3 \log{\left(2 \left(- \frac{1}{10} + \frac{\sqrt{2}}{32}\right) \right)}\right) < 0$$

        /      ___\        /       ___\        /      ___\             
        |2   \/ 2 |        |1    \/ 2 |        |1   \/ 2 |             
- 12*log|- - -----| + 2*log|-- - -----| + 6*log|- - -----| - 4*pi*I < 0
        \5     8  /        \10     32 /        \5     16 /             
    

Then
$$x < \frac{\sqrt{2}}{32}$$
no execute
the solution of our inequality is:
$$x > \frac{\sqrt{2}}{32}$$

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