let log (M) + log(N) = x then 10 ^ [ log(M) + log (N) ] = 10 ^ x. so simplifying we get

10 ^ log(M) *10 ^ log(N) = M*N = 10 ^ x. further simplification we get

log(MN) = x = log(M)  + log (N)

A simple derivation:

log_b MN = log_b (b^{log_b M} b^{log_b N}) = log_b b^{log_b M + log_b N} = log_b M + log_b N.

Nastassia ---

The other answers by Will and Robert are correct.

I would just add an example:

log 12  =  log 4 + log 3  OR  log 6 + log 2, or even log 9 + log (4/3).  Try those on your caluclator.

If you have something like log 3x, it would be log 3 + log x.

If I understand the question, this is just a property of logs.  The log of a product is the sum of the logs. 

log (M * N) = log (M) + log (N)

logMN=logM + logN. This is the log property for production.