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The given function is \( f(x) = \log_2(x) + \log_2(x-2) \), and we are tasked with finding all values of \( x \) for which \( f(x) = 3 \).

### Step-by-Step Solution:

1. **Set up the equation:**
   \[
   f(x) = 3 \quad \Rightarrow \quad \log_2(x) + \log_2(x-2) = 3
   \]

2. **Combine the logarithms:**
   Using the logarithmic identity \( \log_b(a) + \log_b(b) = \log_b(ab) \), we can combine the terms on the left-hand side:
   \[
   \log_2(x) + \log_2(x-2) = \log_2(x(x-2))
   \]
   So, the equation becomes:
   \[
   \log_2(x(x-2)) = 3
   \]

3. **Remove the logarithm:**
   By applying the property \( \log_b(a) = c \) implies \( a = b^c \), we get:
   \[
   x(x-2) = 2^3 = 8
   \]

4. **Simplify the quadratic equation:**
   Expanding the left-hand side:
   \[
   x^2 - 2x = 8
   \]
   Rearranging:
   \[
   x^2 - 2x - 8 = 0
   \]

5. **Solve the quadratic equation:**
   Factor the quadratic equation:
   \[
   (x - 4)(x + 2) = 0
   \]
   So, the possible solutions are:
   \[
   x = 4 \quad \text{or} \quad x = -2
   \]

6. **Check the domain:**
   For the logarithmic expressions \( \log_2(x) \) and \( \log_2(x-2) \), the arguments must be positive. Therefore:
   - \( x > 0 \)
   - \( x - 2 > 0 \) or \( x > 2 \)

   Since \( x = -2 \) does not satisfy these conditions, we discard it.

Thus, the only valid solution is \( x = 4 \).

### Answer:
The correct answer is **B) \( x = 4 \) only**.

Would you like further details on the solution or have any questions about the steps? Here are some related questions:

1. What is the domain of the function \( f(x) = \log_2(x) + \log_2(x-2) \)?
2. How do logarithmic properties help simplify equations like this one?
3. What would happen if \( x \) were a negative value in the logarithmic equation?
4. Can the equation be solved without combining the logarithms?
5. How does one solve quadratic equations using factoring?

**Tip:** Always check the domain restrictions when solving logarithmic equations, as some solutions might be extraneous!

The function f is given by f(x) = log_2(x) + log_2(x - 2). What are all values of x for which f(x) = 3?

Learn how to solve the logarithmic equation f(x) = log_2(x) + log_2(x - 2) and find the values of x where f(x) equals 3. This problem covers logarithmic properties, algebraic manipulation, and quadratic equations. Suitable for students in Grades 9-12.

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