First a little background.  Then the proof:

It comes from the fact that to multiply powers of the same base, you simply
add exponents.

log(a number) is the power to which you must raise 10 to get that number. 

The exponent to which you must raise 10 to get ab is the sum of the power to
which you must raise 10 to get "a" and the power to which you must raise 10 to
get b.



Here's the proof:

Let log(ab) = p, log(a) = q, and log(b) = r.

Then what we want to prove is that p = q+r

Those three logarithm equations by definition of logarithms are equivalent
respectively to these three exponential equations:

10p = ab, 10q = a, 10r = b

Multiply the last two equation (multiply equals by equals):

10q10r = ab

Multiply on the left by adding exponents of 10:

10q+r = ab

But also 10p = ab, so

10p = 10q+r

The bases are the same positive number other than 1, 10.
Therefore the exponents must be equal:

p = q+r

log(ab)= log(a) + log(b)

Edwin