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Let's solve the equation:

\[
\log_2(x - 7) = 4 + \log_2(x + 8)
\]

### Step 1: Move all logarithmic terms to one side.
We can subtract \(\log_2(x + 8)\) from both sides to consolidate the logarithms on one side:

\[
\log_2(x - 7) - \log_2(x + 8) = 4
\]

### Step 2: Apply the properties of logarithms.
Using the property of logarithms \(\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)\), we get:

\[
\log_2\left(\frac{x - 7}{x + 8}\right) = 4
\]

### Step 3: Remove the logarithm.
We can now exponentiate both sides with base 2 to get rid of the logarithm:

\[
\frac{x - 7}{x + 8} = 2^4
\]

\[
\frac{x - 7}{x + 8} = 16
\]

### Step 4: Solve for \(x\).
Now, solve the equation by multiplying both sides by \(x + 8\):

\[
x - 7 = 16(x + 8)
\]

Expanding the right-hand side:

\[
x - 7 = 16x + 128
\]

Now, move all terms involving \(x\) to one side:

\[
x - 16x = 128 + 7
\]

\[
-15x = 135
\]

Solving for \(x\):

\[
x = \frac{135}{-15} = -9
\]

### Step 5: Check for extraneous solutions.
For logarithms to be valid, the arguments inside must be positive:

- \(x - 7 > 0 \Rightarrow x > 7\)
- \(x + 8 > 0 \Rightarrow x > -8\)

The solution \(x = -9\) does not satisfy these conditions, so **there is no solution**.

Would you like any further details?

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Here are five related questions for further exploration:
1. How do logarithmic properties help simplify equations?
2. What are some conditions for a logarithmic equation to have real solutions?
3. What are extraneous solutions, and why do they occur in logarithmic equations?
4. How would the solution change if the logarithms were in a different base?
5. Can you graph the functions involved and see visually where there is no solution?

**Tip:** Always check the domain restrictions when solving logarithmic equations to avoid extraneous solutions.

Solve for x: log2(x - 7) = 4 + log2(x + 8)

Solve the logarithmic equation log2(x - 7) = 4 + log2(x + 8) using properties of logarithms. Learn how to apply logarithmic rules and check for domain restrictions in this detailed step-by-step guide. Suitable for high school students (Grades 10-12).

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