To write \(\log \sqrt[3]{x}\) in expanded form, we can use the properties of logarithms.

Given the expression:

\[ \log \sqrt[3]{x} \]

First, recall that \(\sqrt[3]{x}\) can be written as \(x^{\frac{1}{3}}\).

So, we have:

\[ \log \sqrt[3]{x} = \log x^{\frac{1}{3}} \]

Next, use the power rule of logarithms, which states \(\log a^b = b \log a\):

\[ \log x^{\frac{1}{3}} = \frac{1}{3} \log x \]

Thus, the expanded form of \(\log \sqrt[3]{x}\) is:

\[ \log \sqrt[3]{x} = \frac{1}{3} \log x \]

Would you like more details or have any questions?

Here are some more questions you might have:

1. How do you use other logarithm properties to expand expressions?
2. What are some examples of using the power rule for logarithms?
3. Can you explain the change of base formula for logarithms?
4. How do you simplify logarithmic expressions with different bases?
5. What is the relationship between logarithms and exponents?

**Tip:** Remember that the power rule of logarithms is a powerful tool for simplifying expressions where the argument is raised to a power.

Learn how to expand the logarithm log cube root of x with a step-by-step explanation. This problem involves logarithmic properties and the power rule of logarithms. Suitable for Grades 10-12 students and anyone interested in advanced algebraic concepts.

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