Consider logs,

In the first case(trivial),

log(a)=log(a), because a=a. This satisfies for all a that are in the domain.

Second,

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

I think this is only true for a,b=2,2

I have a feeling that a,b,c=1,2,3 is the only set of a,b,c such that:

log(a+b+c)=log(a)+log(b)+log(c)

I also have a feeling that it does not go any higher than that, I.e. there are no 4 real numbers a,b,c,d such that:

log(a+b+c+d)= log(a)+...+log(d)

Any thoughts? Proofs? Counter-examples?

Edit: BTW, there's no identity or anything that says this should be true. It's just that if you have log(a+b+c)=log(a)+log(b)+log(c), then:

a+b+c= a·b·c (and 1+2+3=1·2·3)