To simplify the expression:

$\ln \left[\frac{x^2 \sqrt{x^3 + 3}}{(x + 3)^7}\right]$

we can apply the properties of logarithms. The key logarithmic rules we'll use are:

1. $\ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b)$
2. $\ln(a \cdot b) = \ln(a) + \ln(b)$
3. $\ln(a^n) = n \ln(a)$

Step 1: Apply the logarithm of a quotient

[ \ln \left[\frac{x^2 \sqrt{x^3 + 3}}{(x + 3)^7}\right] = \ln \left[x^2 \sqrt{x^3 + 3}\right] - \ln \left[(x + 3)^7\right] ]

Step 2: Simplify each term

• For $\ln \left[x^2 \sqrt{x^3 + 3}\right]$, use the product rule $\ln(a \cdot b) = \ln(a) + \ln(b)$:

  $\ln \left[x^2 \sqrt{x^3 + 3}\right] = \ln(x^2) + \ln\left(\sqrt{x^3 + 3}\right)$

  Now, apply the power rule $\ln(a^n) = n\ln(a)$:

  $\ln(x^2) = 2\ln(x)$

  and

  $\ln\left(\sqrt{x^3 + 3}\right) = \frac{1}{2} \ln(x^3 + 3)$

• For $\ln \left[(x + 3)^7\right]$, apply the power rule:

  $\ln \left[(x + 3)^7\right] = 7\ln(x + 3)$

Step 3: Combine everything

Now, we can combine all the simplified terms:

$\ln \left[\frac{x^2 \sqrt{x^3 + 3}}{(x + 3)^7}\right] = 2\ln(x) + \frac{1}{2} \ln(x^3 + 3) - 7\ln(x + 3)$

This is the simplified form of the logarithmic expression.

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Would you like further details on any of the steps, or do you have any questions? Here are five related questions you could explore:

1. How can you apply the logarithmic differentiation to more complex functions?
2. What are other common logarithmic properties useful for simplifying expressions?
3. How do you differentiate functions involving logarithms like $\ln(x^3 + 3)$?
4. What is the general process of simplifying expressions inside logarithms before applying the rules?
5. How does logarithmic expansion work for functions with nested terms like square roots?

Tip: Always remember to handle products and quotients inside logarithms separately using their respective rules for cleaner simplifications.