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Log Inequality Illation (Posted on 2015-03-17) Difficulty: 2 of 5
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.

No Solution Yet Submitted by K Sengupta    
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Solution solution | Comment 1 of 4

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

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