Hi, I am very new to this subject and I don't really understand what's going on here. I understand the basic meaning of the notations, but I don't really get how to compare log and n without log and determine which notation to use. Thanks in advance!

1. (12 marks, 3 marks each] Order notation. For each of the following pairs of functions f(n) and g(n) defined on positive integers, determine the most appropriate symbol in the set {0,0,0} to fill in the blank in the statement f(n) E _(g(n)), if one of these symbols applies. (It is possible that none of the choices applies.) Most appropriate means that you should not answer O if you could answer o or O. Justify your answers. You may use the following facts (based on Skiena, p. 56), where f(n) g(n) is shorthand for f(n) E 0(g(n): 1 log log n log n log? n vnn n log nna 2n! Furthermore, na E O(nb) for ( 0, and n E 0(21). (a) f(n) = 3.14159n3 + 4000n, g(n) = .0001n3 2210 n2; (b) f(n) = log(n!), g(n) = n(log n)?; (c) f(n) = (log n)log logn, g(n) = 2Vlog n; (d) f(n) = (n mod 3)n, g(n) = 0.1n. Here (n mod 3) {0, 1, 2} is the remainder of n when divided by 3. = = =