# Exponential

In algebra, the term "exponential" usually refers to an exponential function. It may also be used to refer to a function that exhibits exponential growth or exponential decay, among other things.

## Exponential function

An exponential function is a function that grows or decays at a rate that is proportional to its current value. It takes the form of

f(x) = b^{x}

where b is a value greater than 0. The rate of growth of an exponential function is directly proportional to the value of the function. There are a few different cases of the exponential function.

### when b = 1

When b = 1 the graph of the function f(x) = 1^{x} is just a horizontal line at y = 1. This is because 1 raised to any power is still equal to 1.

### when b > 1

For f(x) = b^{x}, when b > 1, the graph of the exponential function increases rapidly towards infinity for positive x values. For negative x values, the graph of f(x) approaches 0, but never reaches 0. There is a horizontal asymptote at y = 0, meaning that the graph never touches or crosses the x-axis. Below is the graph of the exponential function f(x) = 3^{x}.

The graph above demonstrates the characteristics of an exponential function; an exponential function always crosses the y axis at (0, 1), and passes through a (in this case 3), at x = 1. The key characteristic of an exponential function is how rapidly it grows (or decays). Just as an example, the table below compares the growth of a linear function to that of an exponential one.

x = 1 | x = 2 | x = 3 | x = 4 | x = 5 | |
---|---|---|---|---|---|

3^{x} |
y = 3 | y = 9 | y = 27 | y = 81 | y = 243 |

3x | y = 3 | y = 6 | y = 9 | y = 12 | y = 15 |

In the table above, we can see that while the y value for x = 1 in the functions 3x (linear) and 3^{x} (exponential) are both equal to 3, by x = 5, the y value for the exponential function is already 243, while that for the linear function is only 15.

### when 0 < b < 1

When b is between 0 and 1, rather than increasing exponentially as x approaches infinity, the graph increases exponentially as x approaches negative infinity, and approaches 0 as x approaches infinity. Below is the graph of .

The graph of the exponential function for values of b between 0 and 1 shares the same characteristics as exponential functions where b > 0 in that the function is always greater than 0, crosses the y axis at (0, 1), and is equal to b at x = 1 (in the graph above (1, ⅓)). Compared to the shape of the graph for b values > 1, the shape of the graph above is a reflection across the y-axis, making it a decreasing function as x approaches infinity rather than an increasing one.

## Natural exponential function

The natural exponential function is f(x) = e^{x}. Since any exponential function can be written in the form of e^{x} such that

b^{x} = e^{x ln(b)}

e^{x} is sometimes simply referred to as the exponential function. It can also be denoted as f(x) = exp(x). As an example, exp(2) = e^{2}. Like the exponential functions shown above for positive b values, e^{x} increases rapidly as x increases, crosses the y-axis at (0, 1), never crosses the x-axis, and approaches 0 as x approaches negative infinity.

One important property of the natural exponential function is that the slope the line tangent to the graph of e^{x} at any given point is equal to its value at that point. In other words, the rate of change of the graph of e^{x} is equal to the value of the graph at that point.

Example

For f(x) = e^{x}, and slope m:

f(0) = e^{0} = 1 = m

f(1) = e^{1} = e = m

f(½) = e^{½} = 1.649 = m

In calculus, this is apparent when taking the derivative of e^{x}. Functions of the form f(x) = ae^{x}, where a is a real number, are the only functions where the derivative of the function is equal to the original function.

The area under the curve (also a topic encountered in calculus) of e^{x} is also equal to the value of e^{x} at x.