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.