The function e^x considered as a real valued function of real numbers is one to one from (-oo, oo) onto (0, oo)
Therefore the real valued logarithm is a well defined function from (0, oo) onto (-oo, oo)
The function e^z considered as a complex valued function of complex numbers is many to one from CC onto CC "\" { 0 }.
In particular, note that e^(2pii) = 1. So if e^z = c then e^(z+2npii) = c for any integer n.
Therefore the complex logarithm has multiple branches, with a principal branch from CC "\" { 0 } onto { x+yi in CC : y in (-pi, pi] } (using the normal range of Arg(z) in (-pi, pi])
The expression ln(z) denotes this principal value.
So whereas z = 7ipi is a root of e^z = -1, it is not the principal value of ln(i^2) = ln(-1).
The principal value is ln(-1) = pii
In general, we can write a formula for the principal value of the logarithm of a complex number z as:
ln z = ln abs(z) + Arg(z) i