The expression given is:

\[
\ln \left( \sqrt[4]{e} \right)
\]

To evaluate this, we can rewrite the fourth root of \( e \) as a power of \( e \):

\[
\sqrt[4]{e} = e^{1/4}
\]

Then, we apply the natural logarithm:

\[
\ln \left( e^{1/4} \right)
\]

Using the logarithmic identity \( \ln (e^x) = x \), we get:

\[
\ln \left( e^{1/4} \right) = \frac{1}{4}
\]

### Final Answer:
\[
\ln \left( \sqrt[4]{e} \right) = \frac{1}{4}
\]

Would you like more details on logarithmic properties, or do you have any other questions?

---

1. What is the logarithmic identity used in the solution?
2. How does changing the base affect logarithmic expressions?
3. How would this problem change if we used a square root instead of a fourth root?
4. How can the natural log function be applied to different powers of \(e\)?
5. What are the applications of logarithms in calculus?

**Tip:** Remember that \(\ln(e^x) = x\) is a fundamental identity and is helpful for simplifying expressions involving \(e\).

This guide provides a solution to evaluating the natural logarithm of the fourth root of e. It uses key logarithmic properties, including \( \ln(e^x) = x \) and the concept of exponents. Perfect for students in Grades 10-12 learning about logarithmic identities.

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