Logarithmic function

The logarithmic is the counterpart to the exponential function. Its domain is defined for all real numbers greater than 0 (a value less than 0 will produce a complex result), and the range is defined for all real numbers.

In the complex case:

$$\ln z=u+iv={\log }_e(|z|)+i(\arg z+2n\pi );n\in \mathbb{Z}$$

Since it has infinitely many values, we sometimes take the principal argument of the complex number and create the principal branch of the logarithmic:

$$\text{Ln}z={\log }_e(|z|)+i\text{Arg}z$$

Even in the complex case, for negative real components the function fails to be analytic.