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We are given the equation:

\[
\log(x + 2) + \log(x - 2) = \log 5
\]

### Step-by-Step Solution:

1. **Combine the logarithms on the left-hand side** using the logarithmic property:

   \[
   \log a + \log b = \log(ab)
   \]

   So, the left-hand side becomes:

   \[
   \log((x + 2)(x - 2)) = \log 5
   \]

2. **Simplify** the expression inside the logarithm:

   \[
   (x + 2)(x - 2) = x^2 - 4
   \]

   Thus, the equation becomes:

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

3. **Since the logarithms are equal**, we can drop the logarithms and set the arguments equal to each other:

   \[
   x^2 - 4 = 5
   \]

4. **Solve for \(x\)**:

   \[
   x^2 = 5 + 4 = 9
   \]

   \[
   x = \pm 3
   \]

### Step 5: Check for Valid Solutions

- If \(x = 3\):
  \[
  x + 2 = 3 + 2 = 5, \quad x - 2 = 3 - 2 = 1
  \]
  Both are positive, so \(x = 3\) is a valid solution.

- If \(x = -3\):
  \[
  x + 2 = -3 + 2 = -1, \quad x - 2 = -3 - 2 = -5
  \]
  Both terms inside the logarithms are negative, which is not allowed in real logarithms.

Therefore, **the only valid solution is**:

\[
x = 3
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you apply logarithmic properties to combine or split terms?
2. What are the conditions under which logarithms are defined?
3. How do we solve quadratic equations like \(x^2 - 4 = 5\)?
4. How can you check if logarithmic solutions are valid?
5. How would this problem change if the logarithms were natural logs?

**Tip**: Always check that the arguments of logarithmic functions are positive when solving equations!

Solve the equation: log(x + 2) + log(x - 2) = log 5

Step-by-step solution to the logarithmic equation log(x + 2) + log(x - 2) = log 5. Learn how to use logarithmic properties to simplify and solve the equation. Suitable for students in Grades 9-12.

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