We are given the logarithmic equation:

\[
\log_{25}(625) = x
\]

This means we need to solve for \(x\) in the equation:

\[
25^x = 625
\]

### Step 1: Express 25 and 625 as powers of 5
We know that:

\[
25 = 5^2 \quad \text{and} \quad 625 = 5^4
\]

So, we can rewrite the equation as:

\[
(5^2)^x = 5^4
\]

### Step 2: Simplify the equation
By the power of a power rule \((a^m)^n = a^{m \cdot n}\), we get:

\[
5^{2x} = 5^4
\]

### Step 3: Set the exponents equal
Since the bases are the same (both are 5), we can set the exponents equal to each other:

\[
2x = 4
\]

### Step 4: Solve for \(x\)
Now, divide both sides by 2:

\[
x = \frac{4}{2} = 2
\]

Thus, the solution is:

\[
x = 2
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you convert logarithmic equations into exponential form?
2. What are the properties of logarithms that can simplify complex expressions?
3. How do you solve logarithmic equations with different bases?
4. How do logarithmic scales work in real-world applications like earthquakes and sound intensity?
5. What is the relationship between logarithms and exponents?

**Tip**: When solving logarithmic equations, always try to express numbers as powers of a common base. This simplifies solving for the unknown.

Learn how to solve the logarithmic equation \log_{25}(625) = x by converting it to an exponential form. This problem demonstrates how to express numbers as powers of the same base and apply logarithmic and exponential rules. Suitable for students in Grades 9-10.

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