5^(x-2)>5^3 inequation

Given the inequality:
$$5^{x - 2} > 125$$
To solve this inequality, we must first solve the corresponding equation:
$$5^{x - 2} = 125$$
Solve:
Given the equation:
$$5^{x - 2} = 125$$
or
$$5^{x - 2} - 125 = 0$$
or
$$\frac{5^{x}}{25} = 125$$
or
$$5^{x} = 3125$$
- this is the simplest exponential equation
Do replacement
$$v = 5^{x}$$
we get
$$v - 3125 = 0$$
or
$$v - 3125 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 3125$$
do backward replacement
$$5^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(5 \right)}}$$
$$x_{1} = 3125$$
$$x_{1} = 3125$$
This roots
$$x_{1} = 3125$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 3125$$
=
$$\frac{31249}{10}$$
substitute to the expression
$$5^{x - 2} > 125$$
$$5^{-2 + \frac{31249}{10}} > 125$$

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the solution of our inequality is:
$$x < 3125$$

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