To determine whether \( \log(n!) \in \Theta(n \log n) \), let's analyze the asymptotic behavior of \( \log(n!) \) and see if it can be bounded both above and below by \( n \log n \) up to constant factors.

### Step 1: Approximate \( \log(n!) \) Using Stirling's Approximation
Stirling's approximation gives us:
\[
n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n
\]
Taking the logarithm of both sides, we get:
\[
\log(n!) \approx \log(\sqrt{2 \pi n}) + \log \left( \frac{n}{e} \right)^n
\]
Breaking this down further:
\[
\log(n!) \approx \frac{1}{2} \log(2 \pi n) + n \log \left( \frac{n}{e} \right)
\]
This simplifies to:
\[
\log(n!) \approx \frac{1}{2} \log(2 \pi n) + n \log n - n
\]
For large \( n \), the \( n \log n \) term dominates, and we can approximate:
\[
\log(n!) \approx n \log n
\]

### Step 2: Establish Big-Theta Bounds
For \( \log(n!) \) to be in \( \Theta(n \log n) \), it needs to be asymptotically bounded both above and below by \( n \log n \).

1. **Upper Bound**: From Stirling's approximation, we know that \( \log(n!) \approx n \log n \), so there exists a constant \( c_1 \) such that:
   \[
   \log(n!) \leq c_1 n \log n
   \]
   for sufficiently large \( n \).

2. **Lower Bound**: Similarly, \( \log(n!) \approx n \log n \) implies there exists a constant \( c_2 \) such that:
   \[
   \log(n!) \geq c_2 n \log n
   \]
   for sufficiently large \( n \).

Since both bounds hold, we can conclude:
\[
\log(n!) \in \Theta(n \log n)
\]

### Final Answer
Yes, \( \log(n!) \in \Theta(n \log n) \).

---

Would you like further details or have any questions?

Here are some related questions:
1. How does Stirling’s approximation work, and why is it useful?
2. How can we apply asymptotic notation for other factorial functions?
3. What other functions can be bounded with \( \Theta(n \log n) \)?
4. How do logarithmic functions compare with other polynomial functions in asymptotic analysis?
5. Can we use Big-O instead of Big-Theta for similar approximations?

**Tip:** Stirling’s approximation is very helpful in estimating the growth rates of factorials and is a valuable tool in combinatorics and algorithm analysis.

Explore whether log(n!) belongs to Θ(n log n) through detailed asymptotic analysis using Stirling’s approximation. This solution covers key concepts such as logarithmic functions and factorial growth rates, suitable for college-level studies.

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