log0.2x>-1 inequation

Given the inequality:
$$\log{\left(0.2 x \right)} > -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(0.2 x \right)} = -1$$
Solve:
Given the equation
$$\log{\left(0.2 x \right)} = -1$$
$$\log{\left(0.2 x \right)} = -1$$
This equation is of the form:

log(v)=p

By definition log

v=e^p

then
$$0.2 x = e^{- 1^{-1}}$$
simplify
$$0.2 x = e^{-1}$$
$$x = \frac{5}{e}$$
$$x_{1} = 1.83939720585721$$
$$x_{1} = 1.83939720585721$$
This roots
$$x_{1} = 1.83939720585721$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1.83939720585721$$
=
$$1.73939720585721$$
substitute to the expression
$$\log{\left(0.2 x \right)} > -1$$
$$\log{\left(0.2 \cdot 1.73939720585721 \right)} > -1$$

-1.05589929264498 > -1

Then
$$x < 1.83939720585721$$
no execute
the solution of our inequality is:
$$x > 1.83939720585721$$

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