For each pair of functions determine if f(n) ? ?(g(n)) or f(n) ? ?(g(n)) or f(n) ? O(g(n)) and provide a proof as specified.

For each of the following, give a proof using the definitions.

1. f(n) = log(n), g(n) = log(n + 1)

2. f(n) = n³ + nlog(n) ? n, g(n) = n⁴ + n

3. f(n) = log(n!), g(n) = nlog(n)

4. f(n) = log₃(n), g(n) = log²(n)

5. f(n) = log(n), g(n) = log(log(n))