0,2^x>=5 inequation

Given the inequality:
$$\left(\frac{1}{5}\right)^{x} \geq 5$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{1}{5}\right)^{x} = 5$$
Solve:
Given the equation:
$$\left(\frac{1}{5}\right)^{x} = 5$$
or
$$-5 + \left(\frac{1}{5}\right)^{x} = 0$$
or
$$\left(\frac{1}{5}\right)^{x} = 5$$
or
$$\left(\frac{1}{5}\right)^{x} = 5$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{5}\right)^{x}$$
we get
$$v - 5 = 0$$
or
$$v - 5 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 5$$
do backward replacement
$$\left(\frac{1}{5}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(5 \right)}}$$
$$x_{1} = 5$$
$$x_{1} = 5$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 5$$
=
$$\frac{49}{10}$$
substitute to the expression
$$\left(\frac{1}{5}\right)^{x} \geq 5$$
$$\left(\frac{1}{5}\right)^{\frac{49}{10}} \geq 5$$

10___     
\/ 5      
----- >= 5
 3125     
     

but

10___    
\/ 5     
----- < 5
 3125    
    

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

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