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**