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To rewrite the given equation without logarithms, we'll need to apply logarithmic properties and exponentiate both sides.

The given equation is:

\[
\ln(x - 8) + \ln(x + 1) = \ln(x - 24)
\]

### Step 1: Apply the logarithmic property for addition:
\[
\ln(a) + \ln(b) = \ln(a \cdot b)
\]
Using this on the left-hand side of the equation, we get:
\[
\ln((x - 8)(x + 1)) = \ln(x - 24)
\]

### Step 2: Remove the logarithms by exponentiating both sides.
Since \(\ln A = \ln B\) implies \(A = B\), we can write:
\[
(x - 8)(x + 1) = x - 24
\]

This is the equation without logarithms. You are now left with a quadratic equation that you can solve later.

Would you like me to help solve for \(x\) or provide any further clarification?

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Here are 5 related questions you might consider next:
1. How do you expand and simplify the quadratic equation \((x - 8)(x + 1)\)?
2. How can you check which values of \(x\) are in the domain of the original logarithmic expressions?
3. What are the key steps in solving quadratic equations?
4. How would the equation change if one of the logarithmic expressions had a different base (like \(\log_{10}\))?
5. What is the significance of ensuring that values of \(x\) don't make the logarithms undefined?

**Tip:** Always check the domain when dealing with logarithmic equations. The argument inside the logarithm must always be positive.

Solve the logarithmic equation ln(x - 8) + ln(x + 1) = ln(x - 24). Rewrite the given equation without logarithms. Do not solve for x.

Learn how to solve the logarithmic equation ln(x - 8) + ln(x + 1) = ln(x - 24). This problem covers key algebraic and logarithmic concepts, including the logarithmic addition property and quadratic equations. Suitable for students in Grades 9-12.

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