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To solve the problem, we need to **expand the given logarithmic expression** \( \ln(x^3 y^2) \) using the properties of logarithms.

### Step-by-Step Solution:

1. **Product Rule of Logarithms**:  
   The logarithm of a product can be written as the sum of the logarithms of the factors. This gives us:
   \[
   \ln(x^3 y^2) = \ln(x^3) + \ln(y^2)
   \]

2. **Power Rule of Logarithms**:  
   The logarithm of a power can be written by bringing the exponent in front of the logarithm. Applying this rule to both terms:
   \[
   \ln(x^3) + \ln(y^2) = 3\ln(x) + 2\ln(y)
   \]

Thus, the expanded form of \( \ln(x^3 y^2) \) is:
\[
3\ln(x) + 2\ln(y)
\]

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Would you like further details on this, or have any additional questions?  
Here are some related questions to extend your understanding:

1. How would you expand \( \ln(a^4 b^5) \)?
2. Can you explain how the quotient rule for logarithms works?
3. What is the expanded form of \( \ln\left( \frac{x^2}{y^3} \right) \)?
4. How does the base of the logarithm affect expansion and simplification?
5. How would you approach solving \( \ln(e^x) \)?

**Tip:** Always remember that logarithmic properties like product, quotient, and power rules make it easier to simplify complex expressions.

Learn how to expand and simplify the logarithmic expression ln(x^3y^2) using the product and power rules of logarithms. This problem involves key algebraic concepts and logarithmic properties, suitable for high school students (Grades 10-12).

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