Let's simplify the given logarithmic expression step by step:

Expression:

$3\ln(x^3 y) + 2\ln(yz^2)$

Step 1: Expand the logarithms using logarithmic properties.

Using the property $\ln(a \cdot b) = \ln(a) + \ln(b)$:

$3\ln(x^3 y) = 3[\ln(x^3) + \ln(y)]$ $2\ln(yz^2) = 2[\ln(y) + \ln(z^2)]$

So the expression becomes: [ 3[\ln(x^3) + \ln(y)] + 2[\ln(y) + \ln(z^2)] ]

Step 2: Apply the power rule.

Using the property $\ln(a^b) = b \cdot \ln(a)$:

$\ln(x^3) = 3\ln(x), \quad \ln(z^2) = 2\ln(z)$

Substituting these, we get: [ 3[3\ln(x) + \ln(y)] + 2[\ln(y) + 2\ln(z)] ]

Step 3: Distribute the coefficients.

$9\ln(x) + 3\ln(y) + 2\ln(y) + 4\ln(z)$

Step 4: Combine like terms.

$9\ln(x) + (3\ln(y) + 2\ln(y)) + 4\ln(z)$ $9\ln(x) + 5\ln(y) + 4\ln(z)$

Final Simplified Expression:

$9\ln(x) + 5\ln(y) + 4\ln(z)$

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Would you like more details or have any questions?
Here are 5 related questions to consider:

1. What are the properties of logarithms and how do they apply?
2. How can logarithmic expressions be simplified for calculus problems?
3. How does the power rule for logarithms help in expansions?
4. Why is it useful to combine like terms in logarithmic expressions?
5. Can you demonstrate these steps for a similar problem, like $2\ln(xy^3) + \ln(yz^2)$?

Tip: Always check if terms in logarithmic expressions can be combined or simplified to reduce complexity before proceeding with further calculations.