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.

We also define special variations:

$${\log }^{(i)}(n)=\{\begin{array}{llc}n& & \text{if and only if}i=0\\ \log ({\log }^{(i-1)}(n))& & \text{if}i\ge 1\end{array}$$

Complexity analysis

For the sake of computational complexity analysis, we also define the iterated logarithm function:

$${\log }^{\ast }n=\min \{i:{\log }^{(i)}n\le 1\}$$

The output denotes how many times the recursive $\log$ function has to be applied to be less than 1. Some key properties:

• This is an extraordinarily short growing function. It is the closest function to $O(1)$ we can possibly get.
• The iterative property, if $n>2$:

$${\log }^{\ast }(a+b)=1+{\log }^{\ast }(\log (n))$$

Complex case

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.