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We are given the equation:

\[
\log_2(x - 7) = 2 + \log_2(x - 4)
\]

We need to solve for \(x\). Let's break it down step by step.

### Step 1: Use logarithmic properties
Since we have logarithms on both sides, we can isolate them. First, subtract \(\log_2(x - 4)\) from both sides:

\[
\log_2(x - 7) - \log_2(x - 4) = 2
\]

### Step 2: Apply the logarithmic subtraction rule
The subtraction rule for logarithms states:

\[
\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)
\]

So, applying this rule:

\[
\log_2\left(\frac{x - 7}{x - 4}\right) = 2
\]

### Step 3: Convert the logarithmic equation to exponential form
Now, rewrite the equation in its exponential form. The logarithmic equation \(\log_b(A) = C\) is equivalent to \(A = b^C\). So:

\[
\frac{x - 7}{x - 4} = 2^2
\]

This simplifies to:

\[
\frac{x - 7}{x - 4} = 4
\]

### Step 4: Solve for \(x\)
Now, multiply both sides by \((x - 4)\) to eliminate the fraction:

\[
x - 7 = 4(x - 4)
\]

Expand the right-hand side:

\[
x - 7 = 4x - 16
\]

Now, solve for \(x\) by bringing all \(x\)-terms to one side:

\[
x - 4x = -16 + 7
\]

This simplifies to:

\[
-3x = -9
\]

Divide both sides by \(-3\):

\[
x = 3
\]

### Step 5: Check the solution
We must check if \(x = 3\) satisfies the original logarithmic conditions. Since the arguments inside the logarithms must be positive:

- \(x - 7 = 3 - 7 = -4\) (which is negative)

Since \(\log_2(x - 7)\) is undefined for negative numbers, \(x = 3\) is **not** a valid solution.

### Final Answer:
There is **no solution** to this equation.

---

Would you like further clarification or details? Here are 5 related questions to explore:

1. How do you convert logarithmic equations to exponential form?
2. What are the key logarithmic properties used in solving such equations?
3. How can you determine the domain of a logarithmic function?
4. Why are negative numbers excluded from the arguments of logarithms?
5. Can you apply logarithmic transformations to both sides of any equation?

**Tip:** Always check for restrictions in the domain when solving logarithmic equations.

Solve for x: log_2(x - 7) = 2 + log_2(x - 4)

Learn how to solve the logarithmic equation log_2(x - 7) = 2 + log_2(x - 4) step by step. This solution covers key logarithmic properties and the conversion to exponential form. Suitable for high school students in Grades 10-12.

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