log4x<=-1 inequation

Given the inequality:
$$\log{\left(4 x \right)} \leq -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(4 x \right)} = -1$$
Solve:
Given the equation
$$\log{\left(4 x \right)} = -1$$
$$\log{\left(4 x \right)} = -1$$
This equation is of the form:

log(v)=p

By definition log

v=e^p

then
$$4 x = e^{- 1^{-1}}$$
simplify
$$4 x = e^{-1}$$
$$x = \frac{1}{4 e}$$
$$x_{1} = \frac{1}{4 e}$$
$$x_{1} = \frac{1}{4 e}$$
This roots
$$x_{1} = \frac{1}{4 e}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{1}{4 e^{1}}$$
=
$$- \frac{1}{10} + \frac{1}{4 e}$$
substitute to the expression
$$\log{\left(4 x \right)} \leq -1$$
$$\log{\left(4 \left(- \frac{1}{10} + \frac{1}{4 e^{1}}\right) \right)} \leq -1$$

          /2    -1\      
pi*I + log|- - e  | <= -1
          \5      /      

Then
$$x \leq \frac{1}{4 e}$$
no execute
the solution of our inequality is:
$$x \geq \frac{1}{4 e}$$

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