log0.2x<=0 inequation

Given the inequality:
$$\log{\left(0.2 x \right)} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(0.2 x \right)} = 0$$
Solve:
Given the equation
$$\log{\left(0.2 x \right)} = 0$$
$$\log{\left(0.2 x \right)} = 0$$
This equation is of the form:

log(v)=p

By definition log

v=e^p

then
$$0.2 x = e^{\frac{0}{1}}$$
simplify
$$0.2 x = 1$$
$$x = 5$$
$$x_{1} = 5$$
$$x_{1} = 5$$
This roots
$$x_{1} = 5$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 5$$
=
$$4.9$$
substitute to the expression
$$\log{\left(0.2 x \right)} \leq 0$$
$$\log{\left(0.2 \cdot 4.9 \right)} \leq 0$$

-0.0202027073175194 <= 0

the solution of our inequality is:
$$x \leq 5$$

 _____          
      \    
-------•-------
       x1