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The function you're dealing with is \( f(x) = x^2 \ln(2x) \), and you're being asked to determine the domain of this function.

### Analyzing the Domain:
- The expression \( x^2 \) is defined for all real values of \( x \), since squaring any real number results in a positive number.
- However, the natural logarithm \( \ln(2x) \) imposes restrictions on the domain. For the logarithm \( \ln(2x) \) to be defined, its argument \( 2x \) must be strictly positive (i.e., \( 2x > 0 \)).
  - This inequality simplifies to \( x > 0 \).

Thus, the domain of \( f(x) = x^2 \ln(2x) \) is restricted to \( x > 0 \).

### Answering the Question:
Based on the analysis:
- **Option A**: The domain is \( (0, \infty) \) in interval notation.
- **Option B**: There are no real values of \( x \) that are excluded except \( x = 0 \). So, the correct answer for the blank is \( x = 0 \).
- **Option C**: The domain is not all real \( x \), because \( x \) must be greater than 0.

So, **Option B** is the correct answer, and you should fill in \( x = 0 \) as the excluded value.

Would you like further details or clarification on any part of this solution?

Here are 5 related questions to expand your understanding:
1. How do you find the domain of a function involving logarithms in general?
2. Why does the function \( \ln(2x) \) require \( 2x > 0 \) to be defined?
3. What happens if \( x = 0 \) is substituted into the original function?
4. Can you explain how transformations of logarithmic functions affect their domains?
5. How would the domain change if the function were \( x^2 \ln(x+1) \) instead?

**Tip**: Always remember that the argument of a logarithmic function must be strictly positive.

Find the domain of f(x) = x^2 ln(2x). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Learn how to find the domain of the function f(x) = x^2 ln(2x), focusing on restrictions imposed by logarithmic expressions. Understand why the argument of the logarithmic function must be positive and how this impacts the domain of the function. Suitable for students in Grades 10-12.

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