exponential function

exponential function   The function f such that f(x) = e^x, or exp x, for all x in R. The two notations arise from different approaches described below, but are used interchangeably. Among the important properties that the exponential function has are the following:
(i) exp(x + y) = (exp x)(exp y), exp (-x) = 1/exp x and (exp x)^r = exp rx. (These hold by the usual rules for indices once the equivalence of exp x and e^x has been established.)
(ii) The exponential function is the inverse function of the logarithmic function: y = exp x if and only if x = ln y.
(iii)
(iv) exp x is the sum of the series
(v) As

Three approaches can be used:
1. Suppose that the value of e has already been obtained independently. Then it is possible to define e^x, the exponential function to base e, by using approach 1 to the exponential function to base a. Then exp x can be taken to mean just e^x. The problem with this approach is its reliance on a prior definition of e and the difficulty of subsequently proving some of the other properties of exp.

2. Define ln as in approach 2 to the logarithmic function, and take exp as its inverse function. It is then possible to define the value of e as exp 1, establish the equivalence of exp x and e^x, and prove the other properties. This is widely held to be the most satisfactory approach mathematically, but it has to be admitted that it is artificial and does not match up with any of the ways in which exp is usually first encountered.
3. Some other property of exp may be used as a definition. It may be defined as the unique function that satisfies the differential equation dy/dx = y (that is, as a function that is equal to its own derivative), with y = 1 when x=0. Alternatively, property (iv) or (v) above could be taken as the definition of exp x. In each case, it has to be shown that the other properties follow .