In the XVIII° century Leonhard Euler extablished the following definitions for the exponential and logarithm functions of a

*real* variable x ...

$\displaystyle e^{x} = \lim_{n \rightarrow \infty} (1+\frac{x}{n})^{n}$ (1)

... and defined the function $\displaystyle \ln x$ as the inverse of the exponential. The alternative to define...

$\displaystyle \ln x = \int_{1}^{x} \frac{dt}{t}$ (2)

... and the exponential the inverse of logarithm has been proposed in the succesive century so that (1) did come first. Apparently each definition approach is reasonably and there are no reason to prefer one of them... but that is not true if we extend the functions to a

*complex* variable z. Ever Euler discovered the famous formula with his name...

$\displaystyle e^{z}= e^{x + i y} = e^{x} (\cos y + i \sin y)$ (3)

... which permits to define the exponential as a

*single value* function for any value of z. Define the logarithm in the complex field as the inverse of exponential is of course possible but if we do that the function $\displaystyle \ln z$ defined from (3) is a

*multivalued function* , i.e. is defined unless a constant $\displaystyle 2 \pi i$ and that is true also if we use the definition...

$\displaystyle \ln z = \int_{1}^{z} \frac{d s}{s}$ (4)

... because the (4) depends from the path connecting the points $\displaystyle s=1$ and $\displaystyle s=z$...

The conclusion, in my opinion, is the following: taking the Euler's way You never will have surprises... taking other ways I don't know

...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$