We are asked to rewrite the expression as a single logarithm. The given expression is:

$\frac{1}{9} \ln((x + 2)^9) + \frac{1}{2} \ln((x^2 + 3x + 2)^2)$

Step 1: Apply the power rule for logarithms.

The power rule states that $a \ln(b) = \ln(b^a)$. Applying this to both terms:

$\ln((x + 2)^9)^{\frac{1}{9}} = \ln((x + 2)^{9 \cdot \frac{1}{9}}) = \ln(x + 2)$ $\ln((x^2 + 3x + 2)^2)^{\frac{1}{2}} = \ln((x^2 + 3x + 2)^{2 \cdot \frac{1}{2}}) = \ln(x^2 + 3x + 2)$

Step 2: Combine using the logarithm addition rule.

The addition rule of logarithms states that $\ln(a) + \ln(b) = \ln(a \cdot b)$. Applying this:

$\ln(x + 2) + \ln(x^2 + 3x + 2) = \ln((x + 2) \cdot (x^2 + 3x + 2))$

Step 3: Simplify the expression.

Now, let's expand $(x + 2)(x^2 + 3x + 2)$:

$(x + 2)(x^2 + 3x + 2) = x(x^2 + 3x + 2) + 2(x^2 + 3x + 2)$ $= x^3 + 3x^2 + 2x + 2x^2 + 6x + 4 = x^3 + 5x^2 + 8x + 4$

Thus, the expression becomes:

$\ln(x^3 + 5x^2 + 8x + 4)$

Final Answer:

The expression rewritten as a single logarithm is:

$\ln(x^3 + 5x^2 + 8x + 4)$

Would you like further details or have any questions?

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Here are 5 related questions:

1. How would the result change if the coefficients in the original expression were different?
2. What happens if we had a subtraction instead of addition between the logarithms?
3. How can logarithmic properties be applied to solve equations involving multiple logarithms?
4. What if the original expression contained natural logarithms with different bases?
5. How can you differentiate the expression obtained after combining the logarithms?

Tip: Remember that logarithmic properties, especially the power and product rules, are key to simplifying complex expressions involving logs!