5^(2*x)>=1/125 inequation

Given the inequality:
$$5^{2 x} \geq \frac{1}{125}$$
To solve this inequality, we must first solve the corresponding equation:
$$5^{2 x} = \frac{1}{125}$$
Solve:
Given the equation:
$$5^{2 x} = \frac{1}{125}$$
or
$$5^{2 x} - \frac{1}{125} = 0$$
or
$$25^{x} = \frac{1}{125}$$
or
$$25^{x} = \frac{1}{125}$$
- this is the simplest exponential equation
Do replacement
$$v = 25^{x}$$
we get
$$v - \frac{1}{125} = 0$$
or
$$v - \frac{1}{125} = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = \frac{1}{125}$$
do backward replacement
$$25^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(25 \right)}}$$
$$x_{1} = \frac{1}{125}$$
$$x_{1} = \frac{1}{125}$$
This roots
$$x_{1} = \frac{1}{125}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{1}{125}$$
=
$$- \frac{23}{250}$$
substitute to the expression
$$5^{2 x} \geq \frac{1}{125}$$
$$5^{\frac{\left(-23\right) 2}{250}} \geq \frac{1}{125}$$

 102         
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 125         
5    >= 1/125
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 5           
         

the solution of our inequality is:
$$x \leq \frac{1}{125}$$

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