The expression you want to evaluate is:

$\log_6 \left( \sqrt[4]{6} \right)$

Let's break this down step by step:

1. The expression inside the logarithm is $\sqrt[4]{6}$, which can be rewritten as $6^{\frac{1}{4}}$.

2. Now, we substitute $\sqrt[4]{6}$ with $6^{\frac{1}{4}}$, so the expression becomes: $\log_6 \left( 6^{\frac{1}{4}} \right)$

3. Using the logarithmic identity $\log_b(b^x) = x$, where $b$ is the base of the logarithm, the result simplifies to: $\frac{1}{4}$

Thus, the value of the logarithmic expression is:

$\log_6 \left( \sqrt[4]{6} \right) = \frac{1}{4}$

Would you like further clarification or more details on logarithmic properties? Here are some related questions for further exploration:

1. How do you simplify $\log_b(b^x)$ in general?
2. What is the general formula for expressing radicals as exponents?
3. How does changing the base of a logarithm affect the evaluation of logarithmic expressions?
4. Can you apply the same principle to $\log_6(\sqrt{6})$? What would the result be?
5. How do you evaluate $\log_6 \left( \frac{1}{6} \right)$?

Tip: Whenever dealing with logarithms, try to rewrite expressions in exponential form, which often simplifies the calculation significantly.