Exponentiation of Real Numbers

General Exponential Functions

Definition: General Exponential Functions

A general exponential function is a complex function $f:\mathbb{C}\to \mathbb{C}$ for which there exists some real number $b>0$ such that $f$ can be expressed using the complex exponential function and the real natural logarithm as follows:

$$f(x)={\mathrm{e}}^{x\ln b}$$

NOTATION

Since ${\mathrm{e}}^{x\ln b}=({\mathrm{e}}^{\ln b}{)}^x$, we write just $b^x$ instead of $f(x)$.