(5x-2)*log(1/3x)<0 inequation

Given the inequality:
$$\left(5 x - 2\right) \log{\left(\frac{x}{3} \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(5 x - 2\right) \log{\left(\frac{x}{3} \right)} = 0$$
Solve:
$$x_{1} = \frac{2}{5}$$
$$x_{2} = 3$$
$$x_{1} = \frac{2}{5}$$
$$x_{2} = 3$$
This roots
$$x_{1} = \frac{2}{5}$$
$$x_{2} = 3$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{2}{5}$$
=
$$\frac{3}{10}$$
substitute to the expression
$$\left(5 x - 2\right) \log{\left(\frac{x}{3} \right)} < 0$$
$$\left(\left(-1\right) 2 + 5 \cdot \frac{3}{10}\right) \log{\left(\frac{1}{3} \cdot \frac{3}{10} \right)} < 0$$

log(10)    
------- < 0
   2       

but

log(10)    
------- > 0
   2       

Then
$$x < \frac{2}{5}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{2}{5} \wedge x < 3$$

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        /     \  
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       x_1      x_2