log0.2*x>=0 inequation

Given the inequality:
$$\log{\left(0.2 x \right)} \geq 0$$
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
$$\log{\left(0.2 x \right)} \geq 0$$
$$\log{\left(0.2 \cdot 4.9 \right)} \geq 0$$

-0.0202027073175194 >= 0

but

-0.0202027073175194 < 0

Then
$$x \leq 5$$
no execute
the solution of our inequality is:
$$x \geq 5$$

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