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!