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.