Tetration

To create tetration, it is better that we work with a commutative exponentiation

$$x\wedge y:=x^{\ln y}.$$

With this definition, tetration would be the solution to the functional equation

$$f(ab)=f(a)\wedge f(b)={f(a)}^{\ln f(b)}$$

Properties

1. $f(1)=e$

2. 

$$\ln f(\frac{1}{x})=\frac{1}{\ln f(x)},$$

so

3. $\ln f(-1)=\pm 1$