A natural logarithm is the inverse of the Real exponential function #x -> e^x# or the Complex exponential function #z -> e^z#.

The domain of #e^x# is the whole of #RR#, while its range is #(0, oo)#. The exponential function is one-one on its domain, so its inverse - the Real natural logarithm, #ln x# - is well defined, with domain #(0, oo)# and range #RR#.

The Complex logarithm is somewhat messier. #z -> e^z# is many to one since #e^(2pii) = 1#. So we need to restrict the domain of #e^z# or the range of #ln z# in order to be able to define #ln z# as a function.

We can define the principal natural logarithm of a Complex number as follows:

#ln z = ln abs(z) + "Arg"(z) i#

The definition of (the principal value of) #"Arg"(z)# may be in the range #(-pi, pi]# or #[0, 2pi)#, resulting in different possible definitions for the principal Complex logarithm.

The one number for which we cannot define a meaningful logarithm is #0#.