Basics of exponential functions

An exponential function is a function of the form y = a^x where a belongs to positive real numbers and x is any real number. We shall first try to understand such numbers with the help of graphs.

Exponential function graph:
Let us try to graph exponential function y = 2^x. So here we see that a = 2 (which is a positive real number and x is any real number). To obtain certain number of points on the graph we construct the following table:

Note that both the above functions are not inverse of each other. With this understanding, let us now define an exponential function.

Definition: Let a belongs to R+. Then the function f: R ->R+, f(x) = a^x is called an exponential function. a is the base of the function. The corresponding exponential function equation would be: y = a^x
For example:
f(x) = 3^x, g(x) = (1/4)^x (where x belongs to R), h(x) = 1^x, (where x belongs to R), are all exponential functions.

Natural exponential function:
We are familiar with the irrational number pi which we come across in connection with the area and circumference of a circle. There is another important irrational number, which is denoted by e and which lies between 2 and 3. Its approximate value is 2.71828. Exact value of e is given by the sequence: e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + ....An exponential function to the base e, f:R->R+, f(x) = e^x is called the natural exponential function. This function is very often used in study of various branches of science and math.

Inverse of exponential function:
Inverse of an exponential function is a logarithmic function. In other words, the exponential function and the logarithmic function are inverses of each other. Thus, if f:R->R+, f(x) = a^x, a belongs to R+ -{1}, then f^(-1):R+->R, f^(-1)(x) = log(a)x [read that as log of x base a]

Derivative of exponential function:
Based on limit definition of derivatives, the derivative of an exponential function can be shown as follows:
If y= a^x, then dy/dx = (a^x)Ln(a)