We can simplify the given expression by using logarithmic properties. Here's the given expression:

$\frac{1}{7} \ln((x + 2)^7) + \frac{1}{2} \ln(x) - \ln((x^2 + 3x + 2)^2)$

Step 1: Apply the Power Rule of Logarithms

The power rule states that $a \ln(b^c) = c \ln(b)$, which allows us to simplify the exponents.

• For $\frac{1}{7} \ln((x+2)^7)$, applying the power rule gives: $\ln(x + 2)$

• For $\frac{1}{2} \ln(x)$, there is no exponent to simplify further, so we leave it as: $\frac{1}{2} \ln(x) = \ln(x^{1/2})$

• For $\ln((x^2 + 3x + 2)^2)$, applying the power rule gives: $2 \ln(x^2 + 3x + 2)$

Now the expression becomes:

$\ln(x + 2) + \ln(x^{1/2}) - 2 \ln(x^2 + 3x + 2)$

Step 2: Use the Product Rule of Logarithms

The product rule states that $\ln(a) + \ln(b) = \ln(ab)$. So we apply this rule to the first two terms:

$\ln((x+2) \cdot x^{1/2})$

This simplifies to:

$\ln((x+2) \sqrt{x})$

Step 3: Use the Quotient Rule of Logarithms

The quotient rule states that $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$. Applying this to the entire expression, we get:

$\ln\left( \frac{(x+2) \sqrt{x}}{(x^2 + 3x + 2)^2} \right)$

Final Answer:

The expression as a single logarithm is:

$\ln\left( \frac{(x+2) \sqrt{x}}{(x^2 + 3x + 2)^2} \right)$

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the power rule of logarithms work in general?
2. Can you explain how the product rule simplifies logarithmic expressions?
3. How would the quotient rule apply if we had more logarithmic terms?
4. What are the domain restrictions for logarithmic expressions like this one?
5. How do the properties of logarithms apply to solving logarithmic equations?

Tip: Always ensure the argument of a logarithm is positive, as logarithms are only defined for positive values.