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:
Logarithm
Exponentiation may be extended to , where is a positive number and the "exponent" is any real number. For example, is the reciprocal of , that is, . Raising "b" to the power 1/2 is the square root of "b". More generally, raising "b" to a rational power "p"/"q", where "p" and "q" are integers, is given by
the "q"-th root of "b". Finally, any irrational number (a real number which is not rational) "y" can be approximated to arbitrary precision by rational numbers. This can be used to compute the "y"-th power of "b": for example formula_9 and formula_10 is increasingly well approximated by formula_11. A more detailed explanation, as well as the formula is contained in the article on exponentiation.