Hence, we have \(\ln(r\cdot\e^{\varphi\cdot\ii})=\ln(r)+ \varphi \cdot\ii\pmod{2k\cdot\pi\cdot\ii}\).

The complex numbers #\ln(r)+ \left(\varphi +2k\cdot\pi\right)\cdot\ii# all have the same image under #\exp#: \[\exp\left(\ln(r)+ \left(\varphi +2k\cdot\pi\right)\cdot\ii\right)=\exp\left(\ln(r)+ \varphi \cdot\ii\right)=r\cdot\e^{\varphi\cdot\ii}\tiny.\]

One often talks about the logarithm #\ln(z)# of #z# as a complex number #w# with the property that #\e^w=z#. Just like the argument this is not a function. One then speaks of a polyvalent function, because they can take on multiple values in a single point. But this is a term we will avoid.

Here we choose to define a specific value by using the principal value #\arg(z)# of the argument of #z#. That value is called, just as with the argument (the argument of a complex number, not to be confused with the argument of a function), the principal value of the logarithm.

Sometimes in the mathematical literature you enounter #\rm Log# for the function we have indicated with #\log#, and #\log(z)# is used to indicate any or all complex solutions #w# of the equation #\e^w=z#.

The choice of #\log# as a function is similar to the one of higher power roots of real numbers: there are two numbers #x# with #x^2=3#, which can both be called a root of #3#. (In general, at higher powers, more than #2# roots can occur.) But with #\sqrt{3}# we indicate the positive solution to the equation. This follows from the choice of the domain #x\ge0# for function #x^2#. By comparison: here we have defined the domain for #\exp# as the set of all complex numbers #z# to #-\pi\lt\Im(z)\le\pi#.

Proof: Assume that #z# and #w# are complex numbers with #\exp(z)=\exp(w)#. Then #\e^z=\e^w# applies, and because of the rules of calculation for complex powers, #\e^{zw}=1#. The definition of complex powers of the number of Euler shows that \[\eqs{ \e^{\Re(zw)}&=&\left|\e^{zw}\right|=\left| 1\right|=1\cr \Im(zw)&=&\arg\left(\e^{zw}\right)=0\pmod{2\pi}\cr}\] such that #zw=\Re(zw)+\Im(zw)\cdot\ii=0\pmod{2\pi\cdot\ii}#, or #z=w\pmod{2\pi\cdot\ii}#. If #z# and #w# both lie in the domain #\left\{z\in\mathbb{C}\mid -\pi\lt \Im(z)\le\pi\right\}#, then their imaginary parts will differ less than #2\pi# from each other, so that they must be equal. This proves #z=w#, and with it the injectivity of #\exp# as a function on the given domain.

It is known that each complex number unequal to #0# has a polar form and therefore can be written as #\e^z# for suitable #z# with #\Im(z)# in #\ivoc{-\pi}{\pi}#. This means that the image of #\exp# on the given domain is #\mathbb{C}\setminus\{0\}#.

The fact that #\ln# is the inverse of #\exp# on the given domain, follow from: \[ \begin{array}{rcl}\ln\left(\e^z\right)&=&\ln\left(\left|\e^{z}\right|\right) + \arg\left(\e^z\right)\cdot\ii\\ &&\phantom{uvwxyz}\color{blue}{\text{definition complex logarithm}}\\ &=&\ln\left(\e^{\Re(z)}\right) + \Im(z)\cdot\ii\\ &&\phantom{uvwxyz}\color{blue}{\text{definition complex function }\exp}\\&=&\Re(z) + \Im(z)\cdot\ii\\ &=&z\\ \end {array}\]

The first equality follows from \[ \begin{array}{rcl}\ln\left(z\cdot w\right)&=& \ln\left(\left|z\cdot w\right|\right)+\arg\left(z\cdot w\right)\cdot\ii\\&=& \ln\left(\left|z\right|\right)+\ln\left(\left| w\right|\right)+\arg\left(z\right)\cdot\ii+\arg\left(w\right)\cdot\ii\pmod{2\pi\cdot\ii}\\&=&\ln(z)+\ln(w)\pmod{2\pi\cdot\ii}\end {array}\] The Second by repeated application of the first one with #z^k=z^{k-1}\cdot z# for #k=n,n-1,\ldots,2#.

The third follows from the fact that the logarithm is the inverse of #\exp#.

For #z# with #-\pi\lt \Im(z)\le\pi # the fourth equality also follows from the fact that the logarithm is the inverse of #\exp#. For other values of #z# the same conclusion cannot be draw, because #z# then not lies within the specially chosen domain of #\exp# where the function is injective. However, direct calculation shows that the complex logarithm behaves like the inverse, modulo #2\pi\cdot\ii#:

\[ \begin{array}{rcl}\ln\left(\e^{z}\right)&=& \ln\left(\left|\e^{z}\right|\right)+\arg\left(\e^{z}\right)\cdot\ii\\&&\phantom{uvwxyz}\color{blue}{\text{definition complex logarithm}}\\&=& \ln\left(\e^{\Re(z)}\right)+\left(\Im\left(z\right)\pmod{2\pi}\right)\cdot\ii\\&&\phantom{uvwxyz}\color{blue}{\text{definition complex function }\exp}\\&=&\left(\Re(z)+\Im(z)\cdot\ii\right) \pmod{2\pi\cdot\ii}\\&&\phantom{uvwxyz}\color{blue}{\text{definition real logarithm}}\\&=&z\pmod{2\pi\cdot\ii}\end {array}\]