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In mathematics , an exponential function is a function of the form. As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function that is, its derivative is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b :. This function, also denoted as exp x , is called the "natural exponential function", [1] [2] [3] or simply "the exponential function". The natural exponential is hence denoted by.

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## How to Write an Exponential Function Given a Rate and an Initial Value

To create this article, 19 people, some anonymous, worked to edit and improve it over time. This article has been viewed , times. Learn more Exponential functions can model the rate of change of many situations, including population growth, radioactive decay, bacterial growth, compound interest, and much more. Follow these steps to write an exponential equation if you know the rate at which the function is growing or decaying, and the initial value of the group. To write an exponential function given a rate and an initial value, start by determining the initial value and the rate of interest.

### 1. Definitions: Exponential and Logarithmic Functions

This formula is proved on the page Definition of the Derivative. Using the chain rule , we have. By the chain rule , we obtain:. Necessary cookies are absolutely essential for the website to function properly.

Introduction to Exponential Functions. Learning Objective s. In addition to linear, quadratic, rational, and radical functions, there are exponential functions. Just as in any exponential expression, b is called the base and x is called the exponent.

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