To determine whether \( n^{\log 5} \) is \( \Theta(n^3) \), we need to compare the growth rates of these two functions.

### Step-by-Step Comparison

1. **Expression 1: \( n^{\log 5} \)**  
   The term \( \log 5 \) is the logarithm of 5 in base 10 (or base 2, depending on context, though base 2 is common in theoretical computer science). So, \( n^{\log 5} \) means that the exponent of \( n \) is approximately \( \log 5 \). Numerically, \( \log 5 \approx 2.32 \).

   Thus, \( n^{\log 5} \approx n^{2.32} \).

2. **Expression 2: \( n^3 \)**  
   This is a simple polynomial function where \( n \) is raised to the power of 3.

### Comparison

- We are comparing \( n^{2.32} \) and \( n^3 \). Clearly, \( n^{2.32} \) grows slower than \( n^3 \), because \( 2.32 < 3 \).

### Conclusion

Since \( n^{2.32} \) grows slower than \( n^3 \), the function \( n^{\log 5} \) is **not** \( \Theta(n^3) \). In fact, we can say:

- \( n^{\log 5} = O(n^3) \) (since it grows slower than \( n^3 \)),
- but \( n^{\log 5} \neq \Omega(n^3) \) (since it does not grow as fast as \( n^3 \)).

Therefore, \( n^{\log 5} \) is **not \( \Theta(n^3) \)**.

Would you like further details on complexity comparisons?

Here are five related questions to explore further:

1. What is the meaning of \( \Theta \), \( O \), and \( \Omega \) in asymptotic notation?
2. How can we compare the growth rates of logarithmic and polynomial functions?
3. What are the key differences between \( O \)-notation and \( \Theta \)-notation?
4. How does base conversion in logarithms affect asymptotic growth rates?
5. Can we have functions that are \( O(n^2) \) but not \( \Theta(n^2) \)?

**Tip:** When comparing functions for asymptotic behavior, always check both the upper and lower bounds to establish whether they are truly \( \Theta \).

Determine whether n^log5 is Θ(n³) with this in-depth solution. Explore growth rate comparisons, logarithmic and polynomial functions, and learn how to apply asymptotic notations like Θ, O, and Ω. Suitable for college-level students in theoretical computer science.

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