To analyze whether \( \log n! \in \Theta(n \log n) \), let's break down both expressions and compare their growth rates.

### Step 1: Approximating \( \log n! \) using Stirling's Approximation
A common approach is to approximate \( n! \) using Stirling's approximation:
\[
n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n
\]
Taking the logarithm of both sides gives:
\[
\log(n!) \approx \log \left( \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \right)
\]
This simplifies to:
\[
\log(n!) \approx \log(\sqrt{2 \pi n}) + \log \left( \frac{n}{e} \right)^n
\]
\[
\log(n!) \approx \frac{1}{2} \log(2 \pi n) + n \log n - n
\]
As \( n \) becomes large, the \( n \log n \) term dominates, so:
\[
\log(n!) \approx n \log n
\]

### Step 2: Concluding with Big Theta Notation
Since \( \log(n!) \) grows asymptotically at the same rate as \( n \log n \), we can conclude:
\[
\log(n!) \in \Theta(n \log n)
\]

Thus, \( \log n! \in \Theta(n \log n) \) is correct.

Would you like more details, or have any questions?

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1. How does Stirling's approximation help in asymptotic analysis?
2. What is the difference between Big-O and Theta notation?
3. Why is \( n \log n \) considered a dominant term in this analysis?
4. How would the asymptotic behavior change if we considered a smaller function, like \( \log n \)?
5. What is an intuitive way to understand factorial growth in terms of logarithms?

**Tip:** When analyzing factorial functions, approximations like Stirling's are valuable for simplifying logarithmic growth comparisons.

Explore whether log(n!) belongs to Θ(n log n) by applying Stirling's Approximation for factorials. This problem covers asymptotic analysis, logarithmic growth, and the use of Theta notation, making it suitable for advanced high school or introductory college math students.

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