The given expression is:

\[
\log \left( \sqrt[3]{10x} \right)
\]

To simplify this, we can use the property of logarithms that states:

\[
\log(a^b) = b \log(a)
\]

The cube root of \(10x\) can be written as \((10x)^{1/3}\). Now apply the logarithmic property:

\[
\log \left( (10x)^{1/3} \right) = \frac{1}{3} \log(10x)
\]

Next, using the property of logarithms that allows us to separate the terms:

\[
\log(10x) = \log(10) + \log(x)
\]

Therefore, the expression becomes:

\[
\frac{1}{3} \left( \log(10) + \log(x) \right)
\]

Since \(\log(10) = 1\) (in base 10):

\[
\frac{1}{3} (1 + \log(x)) = \frac{1}{3} + \frac{1}{3} \log(x)
\]

Thus, the simplified form of the expression is:

\[
\frac{1}{3} + \frac{1}{3} \log(x)
\]

Would you like further details or have any questions?

Here are some related questions to explore:
1. How would the expression change if the base of the logarithm was different (e.g., natural logarithm)?
2. Can you simplify \(\log \left( \sqrt{10x} \right)\) instead of a cube root?
3. What if \(x\) was a specific value, say \(x = 100\)? How does this affect the expression?
4. How would you expand \(\log(100x)\) similarly?
5. How do logarithmic properties apply to more complex functions?

**Tip:** Always remember the key properties of logarithms for simplifying expressions, such as product, quotient, and power rules!

This guide walks you through simplifying the logarithmic expression log(∛(10x)) step by step using logarithmic properties. Suitable for students in Grades 9-11, this problem covers key concepts such as the power and product rules of logarithms.

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