Let's expand the given logarithmic expression using properties of logarithms.

The expression is:

$\ln \left( \frac{x^3 \sqrt{x^2 + 2}}{(x + 2)^6} \right)$

Step-by-step Expansion:

1. Quotient Rule of Logarithms: Using the property: $\ln \left( \frac{A}{B} \right) = \ln(A) - \ln(B)$ We apply it to the expression: $\ln \left( \frac{x^3 \sqrt{x^2 + 2}}{(x + 2)^6} \right) = \ln \left( x^3 \sqrt{x^2 + 2} \right) - \ln \left( (x + 2)^6 \right)$

2. Power Rule of Logarithms: For the first term, we use the product rule: $\ln(AB) = \ln(A) + \ln(B)$ This gives: $\ln \left( x^3 \sqrt{x^2 + 2} \right) = \ln(x^3) + \ln(\sqrt{x^2 + 2})$

   Then, apply the power rule: $\ln(x^n) = n \ln(x)$ $\ln(x^3) = 3 \ln(x)$ And for the square root: $\ln(\sqrt{x^2 + 2}) = \ln \left( (x^2 + 2)^{1/2} \right) = \frac{1}{2} \ln(x^2 + 2)$

3. Simplify: Now let's rewrite the expression with these simplifications: $\ln \left( \frac{x^3 \sqrt{x^2 + 2}}{(x + 2)^6} \right) = 3 \ln(x) + \frac{1}{2} \ln(x^2 + 2) - \ln \left( (x + 2)^6 \right)$

   Applying the power rule to the last term: $\ln \left( (x + 2)^6 \right) = 6 \ln(x + 2)$

4. Final Expression: Putting everything together, the expanded form is: $3 \ln(x) + \frac{1}{2} \ln(x^2 + 2) - 6 \ln(x + 2)$

Would you like further clarification or additional details?

Relative Questions:

1. How does the quotient rule simplify complex logarithmic expressions?
2. Can you explain the power rule of logarithms in more depth?
3. How do you apply the product rule for logarithms in multiple cases?
4. What other properties of logarithms are useful for simplifications?
5. How would this expression change if different bases were used?

Tip:

Always break down logarithmic expressions step-by-step using known properties like the power, product, and quotient rules.