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We are given the equation:

\[
\log_2(x) + \log_2(x - 24) = 8
\]

To solve this, follow these steps:

### Step 1: Combine the logarithmic terms
Using the logarithmic property \(\log_b(a) + \log_b(c) = \log_b(a \cdot c)\), we can combine the left-hand side:

\[
\log_2(x) + \log_2(x - 24) = \log_2(x(x - 24))
\]

Now the equation becomes:

\[
\log_2(x(x - 24)) = 8
\]

### Step 2: Eliminate the logarithm
To eliminate the logarithm, rewrite the equation in exponential form. Since \(\log_2(y) = 8\) means \(y = 2^8\), we get:

\[
x(x - 24) = 2^8
\]

\[
x(x - 24) = 256
\]

### Step 3: Solve the quadratic equation
Now expand and solve the quadratic equation:

\[
x^2 - 24x = 256
\]

\[
x^2 - 24x - 256 = 0
\]

### Step 4: Apply the quadratic formula
The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 - 24x - 256 = 0\), the coefficients are \(a = 1\), \(b = -24\), and \(c = -256\). Substituting these values into the quadratic formula:

\[
x = \frac{-(-24) \pm \sqrt{(-24)^2 - 4(1)(-256)}}{2(1)}
\]

\[
x = \frac{24 \pm \sqrt{576 + 1024}}{2}
\]

\[
x = \frac{24 \pm \sqrt{1600}}{2}
\]

\[
x = \frac{24 \pm 40}{2}
\]

### Step 5: Find the solutions
Now calculate the two possible values for \(x\):

1. \(x = \frac{24 + 40}{2} = \frac{64}{2} = 32\)
2. \(x = \frac{24 - 40}{2} = \frac{-16}{2} = -8\)

### Step 6: Check for valid solutions
Since logarithms are only defined for positive arguments, \(x > 0\) and \(x - 24 > 0\). This implies \(x > 24\). Therefore, the solution \(x = -8\) is invalid.

Thus, the only valid solution is:

\[
x = 32
\]

### Final Answer:
\[
x = 32
\]

Would you like further details or clarification on any steps?

Here are 5 related questions:
1. What are the properties of logarithms used in this solution?
2. How do we convert a logarithmic equation to exponential form?
3. Why are logarithms only defined for positive arguments?
4. How is the quadratic formula derived?
5. What would the equation look like if the base of the logarithm were different?

**Tip:** When solving logarithmic equations, always check that the solutions do not result in taking the logarithm of a non-positive number!

Solve for x: log_2(x) + log_2(x - 24) = 8

Learn how to solve the logarithmic equation log_2(x) + log_2(x - 24) = 8 step by step. This problem involves logarithmic properties and solving a quadratic equation, suitable for high school students in Grades 10-12.

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