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The question provided is:

\[
\text{If } \ln(x) + \ln(x - 7) = \ln(8x), \text{ then } x = ?
\]

Let's solve this step-by-step:

1. **Combine the Logarithmic Terms**: Use the property of logarithms that states \(\ln(a) + \ln(b) = \ln(a \cdot b)\).
   \[
   \ln(x) + \ln(x - 7) = \ln(x \cdot (x - 7))
   \]
   This simplifies to:
   \[
   \ln(x(x - 7)) = \ln(8x)
   \]

2. **Rewrite the Equation**: Now we have:
   \[
   \ln(x(x - 7)) = \ln(8x)
   \]

   Since the natural logarithm of both sides is equal, we can set the arguments equal to each other:
   \[
   x(x - 7) = 8x
   \]

3. **Expand and Simplify**: Distribute \(x\) on the left side.
   \[
   x^2 - 7x = 8x
   \]

4. **Move All Terms to One Side**: Subtract \(8x\) from both sides.
   \[
   x^2 - 15x = 0
   \]

5. **Factor the Equation**: Factor out \(x\).
   \[
   x(x - 15) = 0
   \]

6. **Solve for \(x\)**: Set each factor to zero.
   - \(x = 0\)
   - \(x - 15 = 0 \Rightarrow x = 15\)

7. **Check for Valid Solutions**: In the context of logarithmic functions, \(x\) must be positive, and \(x - 7\) must also be positive to ensure the arguments of the logarithms are defined.

   - For \(x = 0\): This does not satisfy the conditions, as \(\ln(0)\) is undefined.
   - For \(x = 15\): This is valid because both \(x\) and \(x - 7\) (i.e., 8) are positive.

Thus, the solution is:
\[
x = 15
\]

Would you like further details on any of these steps, or do you have additional questions?

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### Related Questions
1. What are other properties of logarithms that can simplify equations?
2. How do you handle equations involving logarithms with different bases?
3. What are common mistakes to avoid when solving logarithmic equations?
4. Can you solve similar equations where logarithms are combined differently?
5. How do you check the validity of solutions in logarithmic equations?

### Tip
Always remember to verify that solutions satisfy the domain of the logarithmic function (i.e., the argument of any logarithmic term must be positive).

If ln(x) + ln(x - 7) = ln(8x), then x = ?

Learn how to solve the logarithmic equation ln(x) + ln(x - 7) = ln(8x) using logarithmic properties. This detailed solution covers algebraic concepts, combining logarithmic terms, and setting arguments equal for solutions. Suitable for college-level introductory algebra students.

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