To solve the equation \( \frac{5^x}{25} = \frac{5^x \cdot 5^x}{25} = 5^x \), we can start by simplifying the expressions.

1. **Convert 25 to base 5**:
\[
25 = 5^2
\]

2. **Substitute in the equation**:
\[
\frac{5^x}{25} = \frac{5^x}{5^2} = 5^{x - 2}
\]

The second part of the equation:
\[
\frac{5^x \cdot 5^x}{25} = \frac{(5^x)^2}{25} = \frac{5^{2x}}{25} = \frac{5^{2x}}{5^2} = 5^{2x - 2}
\]

3. **Now the equation looks like this**:
\[
5^{x - 2} = 5^{2x - 2} = 5^x
\]

Now we will set the exponents equal to each other:

4. **Set the exponents equal**:
\[
x - 2 = x \quad \text{(first equation)}
\]
\[
2x - 2 = x \quad \text{(second equation)}
\]

From the first equation \( x - 2 = x \):
- This leads to a contradiction \( -2 = 0 \), which has no solution.

From the second equation \( 2x - 2 = x \):
- Rearranging gives:
\[
2x - x = 2
\]
\[
x = 2
\]

5. **Check if \( x = 2 \) satisfies the original equation**:
Substitute \( x = 2 \):
\[
\frac{5^2}{25} = \frac{25}{25} = 1
\]
\[
5^2 = 25
\]

Both sides are equal, confirming that \( x = 2 \) is indeed a solution.

Thus, the solution to the equation is:
\[
\boxed{2}
\]