In this exercise, you're given two algorithms, A and B, which have time complexities represented by functions $T_A(n)$ and $T_B(n)$, respectively. Specifically:

• $T_A(n) = 0.1n^2 \log_{10} n$ (microseconds) for algorithm A.
• $T_B(n) = 2.5n^2$ (microseconds) for algorithm B.

You need to:

1. Compare the two algorithms in the Big-O notation: Big-O notation helps classify the growth rates of functions, so you're being asked to determine which algorithm grows slower for large $n$, meaning it performs better for large problem sizes.

2. Find a problem size $n_0$ where one algorithm begins to outperform the other**: For $n > n_0$, you want to identify at what point one of the algorithms starts to consistently outperform the other.

3. Give a recommendation: If your problems have sizes $n > 10^9$, which algorithm (A or B) should you choose based on the analysis.

What you need to do:

• First, compare the terms $0.1n^2 \log_{10} n$ and $2.5n^2$ to see how they scale as $n$ increases. Since both have $n^2$, the comparison comes down to how $\log_{10} n$ behaves compared to the constant coefficient in $T_B(n)$.

• Solve for $n_0$ by setting $T_A(n) = T_B(n)$ and solve for $n$ to determine when one algorithm overtakes the other.

• Use the analysis to recommend which algorithm to choose for $n > 10^9$.

Would you like me to walk through these steps in more detail, including solving for $n_0$?