stylistic:tone
Problems with inverting the complex exponential function

For a function to have an inverse, it must map distinct values to distinct values; that is, it must be injective. But the complex exponential function is not injective, because e w + 2 π i k = e w {\displaystyle e^{w+2\pi ik}=e^{w}} for any complex number w {\displaystyle w} and integer k {\displaystyle k}, since adding i θ {\displaystyle i\theta } to z {\displaystyle z} has the effect of rotating e w {\displaystyle e^{w}} counterclockwise θ {\displaystyle \theta } radians. So the points

equally spaced along a vertical line, are all mapped to the same number by the exponential function. This means that the exponential function does not have an inverse function in the standard sense. There are two solutions to this problem.

One is to restrict the domain of the exponential function to a region that does not contain any two numbers differing by an integer multiple of 2 π i {\displaystyle 2{\mathit {\pi i}}}: this leads naturally to the definition of branches of log z {\displaystyle \log z}, which are certain functions that single out one logarithm of each number in their domains. This is analogous to the definition of arcsin x {\displaystyle \arcsin x} on {\displaystyle } as the inverse of the restriction of sin θ {\displaystyle \sin \theta } to the interval {\displaystyle }: there are infinitely many real numbers θ {\displaystyle \theta } with sin θ = x {\displaystyle \sin \theta =x}, but one arbitrarily chooses the one in {\displaystyle }.

Another way to resolve the indeterminacy is to view the logarithm as a function whose domain is not a region in the complex plane, but a Riemann surface that covers the punctured complex plane in an infinite-to-1 way.

Branches have the advantage that they can be evaluated at complex numbers. On the other hand, the function on the Riemann surface is elegant in that it packages together all branches of the logarithm and does not require an arbitrary choice as part of its definition.