Exponential family
What is important to note, and what characterizes all exponential family variants, is that the parameter(s) and the observation variable(s) must factorize (can be separated into products each of which involves only one type of variable), either directly or within either part (the base or exponent) of an exponentiation operation. Generally, this means that all of the factors constituting the density or mass function must be of one of the following forms:
Exponentiation
The exponential function is defined for all integer, fractional, real, and complex values of "x". In fact, the matrix exponential is well-defined for square matrices (in which case this exponential identity only holds when "x" and "y" commute), and is useful for solving systems of linear differential equations.