e is a special mathematical constant approximately equal to 2.71828, discovered by Euler. It is the base of the natural logarithm and is used in exponential functions of the form f(x) = aerx to model growth and decay. The value of e can be evaluated using calculators and exponential functions involving e can be graphed by plotting points and shifting the parent graph as needed.

2. 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.

3. 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.)

4. Simplifying Natural Base
Expressions
 Follow the same exponent rules as with
other bases.
 Examples:




5. Your Turn!
 Simplify

6. Evaluating Natural Base
Expressions
 Use a calculator to evaluate each
expression.
 Press “2nd” then “LN” key to get to ex .
 Examples:
 e2
 e-0.06

7. Natural Base Exponential
Functions
 Functions 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.

8. 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.

9. Examples:
 Graph. Then state the domain and range.

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

11. Continuous Interest
 Remember, compound interest uses the
equation:
 As n approaches +∞ it is called continuously
compounded interest.
 It is then modeled by:

12. Example:
 You deposit $1000 in an account that pays
8% annual interest compounded
continuously.
 What is the balance after 1 year?

13. Your Turn!
 You deposit $1500 in an account that pays
7.5% annual interest compounded
continuously.
 What is the balance after 3 years?