 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/discrete-mathematics-probability-and-continuous-mathematics/2/b/ii/solution

n lo g_{2} (5 n) + lo g_{2} (n^{2}) = n lo g_{2} n + n lo g_{2} 5 + 2 lo g_{2} n .

For

n \geq 2

, both

n \leq n lo g_{2} n

and

lo g_{2} n \leq n lo g_{2} n

. The expression is therefore at most

(3 + lo g_{2} 5) n lo g_{2} n

. Consequently

True: n lo g_{2} (5 n) + lo g_{2} (n^{2}) = O (n lo g_{2} n) .

The Big O notation concerns eventual behaviour; the zero value of

n lo g_{2} n

n = 1

is irrelevant.

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