To simplify the expression \(6\ln(x) + 3\ln(y) - 4\ln(xy)\), we can use properties of logarithms.

1. **Use the property of logarithms:** 
   \(\ln(ab) = \ln(a) + \ln(b)\) and \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\).

2. **Expand \( \ln(xy) \):**
   \[
   \ln(xy) = \ln(x) + \ln(y)
   \]
   Therefore,
   \[
   -4\ln(xy) = -4(\ln(x) + \ln(y)) = -4\ln(x) - 4\ln(y)
   \]

3. **Substitute back into the expression:**
   \[
   6\ln(x) + 3\ln(y) - 4\ln(x) - 4\ln(y)
   \]

4. **Combine like terms:**
   \[
   (6\ln(x) - 4\ln(x)) + (3\ln(y) - 4\ln(y)) = 2\ln(x) - \ln(y)
   \]

Thus, the simplified expression is:
\[
2\ln(x) - \ln(y)
\]

### Summary:
The simplified form of \(6\ln(x) + 3\ln(y) - 4\ln(xy)\) is \(2\ln(x) - \ln(y)\).

Would you like more details or have any questions? Here are some related questions to consider:

1. What are other properties of logarithms?
2. How can you solve for \(x\) or \(y\) if you have specific values?
3. What is the derivative of the simplified expression?
4. How does this simplification apply in real-world scenarios?
5. Can logarithmic identities be applied to different bases? 

**Tip:** Always remember the key properties of logarithms for simplification: product, quotient, and power rules.

Explore the simplification of the logarithmic expression 6ln(x) + 3ln(y) - 4ln(xy) into 2ln(x) - ln(y). This guide covers key logarithmic properties and is suitable for high school students.

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