# The Natural Base: e

## Objectives

1. What is e?
2. When do I use e?
3. Where did e derived from?
4. What shape does f(x)=e^x look like

Everything that grows in life is exponential. Bacteria grows at an exponential rate. Human population grows at an exponential rate. A convenient way to model exponential growth and decay is by using e.

## What is e?

What is e? For starters, e is called the natural base. It is defined as:

e=2.71828...

(Therefore, e is an irrational number like pi. It is NOT a variable).

The number e is the base of the natural logarithm.

## When do I use e?

e is a convenient way to model exponential growth and decay. It can be applied to compound interest and sum of the infinite series (Taylor Series).

(The equation above is continuous compounding.)
(The equation above is the sum of infinite series.)

## Where did e derive from?

e is the value of the derivative of the function:

f(x)= e^x at the point x=0 is equal to 1.

This equation is called the natural exponential function.

(The picture above is a graph of natural exponential function).

## The most important things to remember!

• e is called the natural base
• f(x)= e^x is called the natural exponential function
• e is derived from the natural exponential function
• e is used in applications involving growth and decay (e.g. continuous compound)
• The graph of f(x)= e^x looks like a J