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The equation given is:

\[
\ln \sqrt[7]{\frac{x^2 + 7}{x^3 + 4}} = ?
\]

To solve this, let's proceed step by step:

### Step 1: Apply the logarithmic rule
The general rule for logarithms is:
\[
\ln(a^b) = b \cdot \ln(a)
\]
We can rewrite the equation as:

\[
\ln \sqrt[7]{\frac{x^2 + 7}{x^3 + 4}} = \frac{1}{7} \ln \left( \frac{x^2 + 7}{x^3 + 4} \right)
\]

### Step 2: Use the logarithmic property for division
Using the logarithmic property:
\[
\ln \left( \frac{A}{B} \right) = \ln(A) - \ln(B)
\]
We apply this to our expression:
\[
\frac{1}{7} \ln \left( \frac{x^2 + 7}{x^3 + 4} \right) = \frac{1}{7} \left( \ln(x^2 + 7) - \ln(x^3 + 4) \right)
\]

### Final Answer:
The simplified form of the expression is:
\[
\frac{1}{7} \left( \ln(x^2 + 7) - \ln(x^3 + 4) \right)
\]

Let me know if you'd like further details or have any other questions!

### Related Questions:
1. How do you apply the chain rule when differentiating logarithmic functions?
2. What is the importance of logarithmic identities in simplifying expressions?
3. Can you explain how to solve logarithmic equations involving radicals?
4. How does the property \(\ln \left( \frac{A}{B} \right) = \ln(A) - \ln(B)\) help in solving complex logarithmic expressions?
5. How can we apply logarithmic differentiation to complex functions?

### Tip:
Always remember to simplify expressions before applying logarithmic rules; it can make complex equations much easier to handle.

Solve for x: ln(7th root of ((x^2 + 7) / (x^3 + 4))) = ?

This problem delves into the logarithmic properties and algebraic manipulation of expressions. Learn how to simplify and solve the expression ln(7th root of ((x^2 + 7) / (x^3 + 4))) using basic logarithmic identities and algebraic techniques. Suitable for college-level students.

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