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Evaluate the following logarithmic expressions without using a calculator.

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.

Evaluate the following logarithmic expression: log_6(√[4]{6})

Learn how to evaluate the logarithmic expression log_6(√[4]{6}) by using properties of logarithms and exponents. This step-by-step solution simplifies the radical form and applies key logarithmic identities, suitable for high school students in Grades 9-12.

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