# Exponential function

The **exponential function** is the function , exponentiation by *e*. It is a very important function in analysis, both real and complex.

## General Info and Definitions

Exponential functions are functions that grows or decays at a constant percent rate.

- Exponential functions that result in an
of**increase***y*is called an.**exponential growth** - Exponential functions that result in an
of**decrease***y*is called an.**exponential decay**

An exponential growth graph looks like:

An exponential decay graph looks like:

Exponential functions are in one of three forms.

- , where
*b*is the % change written in decimals - , where e is the irrational constant
*2.71828182846....* - or , where
*h*is the half-life (for decay), or*d*is the doubling time (for growth).

Whether an exponential function shows growth or decay depends upon the value of its *b* value.

- If , then the function will show growth.
- If , then the function will show decay.

## Solving Exponential Equations

There are two ways to solve an exponential equation. Graphically with a computer/calculator or algebraicly using logarithms.

**Example:** Solve

**Graphically:**

**Algebraically:**

There, we will use natural logarithms. The same operation can also be done with common logarithms.