Will the inequality log M ≥ C*log 2 always hold, where M is a positive integer and C is the number of distinct prime numbers that divide M?
If so, prove it.
If not, give an example.

let M=2^a * 3^b *....p^K   (C powers of distinct primes)

 log M=alog2+blog3 +... klogp >log2+log3+... logp >= C*log2

log M = C*log 2  only for M=2^n, otherwise log M > C*log 2

edit:  n can be zero:

It is true for M=1 AS WELL since log1=0 and C=0 

Edited on March 17, 2015, 9:12 am

Posted by Ady TZIDON on 2015-03-17 09:07:59