Prove this using the definition R7: log(n*) is O(log n) for any fixed x > 0

Question

Question

Prove this using the definition

R7: log(n*) is O(log n) for any fixed x > 0

Answer 1

Answer #1

From logarithm property, we know that

log(n^x) = x.log(n)

Now, from Big-O time complexity, we know that:

O(g(n)) = { f(n): there exist positive constants c and 
                  n0 such that 0 <= f(n) <= c*g(n) for 
                  all n >= n0}

Clearly, c >= x and n >= 1, we have

0 <= x.log(n) <= c.log(n)

Hence, log(n^x) = = O(logn)

Answer 2

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