Can I find the natural log of a negative number?

1 Answer
George C.
Nov 5, 2015

Yes, if #x < 0# then the principal value of #ln(x)# is #ln(-x)+i pi#

Explanation:

The Real valued function #e^x:RR -> (0, oo)# is one to one, with inverse function #ln(x):(0, oo)->RR#.

We can extend the definition of #e^x# to the Complex valued function #e^z:CC->CC\{0}#, but this is a many to one function, so it has no inverse function, unless we do something to limit the domain of #e^z# or the range of #ln z#.

For example, if we limit the domain of #e^z# to the set #{a+ib in CC : -pi < b <= pi}#, then it is a one to one function with inverse function:

#ln(z):CC\{0} -> {a+ib in CC : -pi < b <= pi}#

If #x < 0#, then #e^(ln(-x)+i pi) = e^ln(-x) e^(i pi) = -x * -1 = x#

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