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:

1. (i) exp(x+y) = (expx)(expy), exp(−x) = 1/ exp x and (exp x)^r=exp rx.

2. (ii) The exponential function is the inverse function of the logarithmic function: y = expx if and only if x = lny.

3. (iii) $\frac{\text{d}}{\text{d}x}(\text{exp}x)=\text{exp}x$.

4. (iv) exp x is the sum of the series $1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}+\cdots .$

5. (v) As n→∞, ${\left(1+\frac{x}{n}\right)}^n\to \text{exp}x.$

Graph of exponential function

The exponential function may be defined or characterized in different ways.

1. 1. Define ln as in the logarithmic function entry, 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 above.

2. 2. The power series in (iv) converges for all real numbers (and in fact also all complex numbers (see complex exponential)). This can be seen from the ratio test. By applying appropriate theorems of analysis, the other properties can be proved.

3. 3. expx may be characterized as the unique function that satisfies the differential equation dy/dx = y with y(0) = 1.