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