What is e?Much like π and i, e is a special number

used in math.

Discovered by mathematician Leonhard

Euler. (Sounds like “oiler”)

Called the natural base e or the Euler

number.

Investigating e

n 10 100 1000 10,000 100,000 1,000,000

•As n approaches +∞, approaches

e≈2.718281828459

•The natural base e is irrational. (It cannot be

expressed as a fraction.)

Simplifying Natural Base

ExpressionsFollow the same exponent rules as with

other bases.

Examples:

Your Turn! Simplify

Evaluating Natural Base

ExpressionsUse a calculator to evaluate each

expression.

Press “2nd” then “LN” key to get to ex .

Examples:

e2

e-0.06

Natural Base Exponential

FunctionsFunctions of the form f(x) = aerx are called

natural base exponential functions.

If r > 0 it is exponential growth.

If r < 0 it is exponential decay.

Graphing Natural Base Functions

Plot points (0, a) and (1, ___)

If points are too close together, you may

choose a different x for the 2nd point.

Shift parent graph using h and k if needed.

Examples:Graph. Then state the domain and range.

Your Turn!Graph. Then state the domain and range.

Continuous InterestRemember, compound interest uses the

equation:

As n approaches +∞ it is called continuously

compounded interest.

It is then modeled by:

Example:You deposit $1000 in an account that pays

8% annual interest compounded

continuously.

What is the balance after 1 year?

Your Turn!You deposit $1500 in an account that pays

7.5% annual interest compounded

continuously.

What is the balance after 3 years?