Yes, it's to do with complex numbers.
Any complex number can be written as
z=∣z∣eiargz therefore, taking logs means
logz=log(∣z∣eiargz)=log∣z∣+log(eiargz)=log∣z∣+iargzBut since
argz can take on multiple values (we call them
branches) then we can agree on the so-called principal value of
logz. This occurs when we only take the argument of
z to be over the interval
−π<argz≤πHence, we have the natural log of a complex number
z along the principal branch as:
Log z=log∣z∣+iArg zwhere the capital L and A letters symbolise that we are taking the principal branch value.
Anyway, once we have this established, it is clear that for
z=−2 we have
Log (−2)=2−2πiIf you want other branches, just add/subtract
2π from the argument; hence really, it means that
log(−2) takes on infinitely many values of the form
log(−2)=2−(2π+2πn)iwhere
n∈Z.