1. Chapter 3 Exponential and Logarithmic Functions
Pre-Calculus
Chapter 3
Exponential and Logarithmic Functions

2. 3.1 Exponential Functions
Objectives:
Recognize and evaluate exponential functions with base a.
Graph exponential functions.
Recognize, evaluate, and graph exponential functions with base e.
Use exponential functions to model and solve real-life problems.

3. Definition of Exponential Function
The exponential function with base a is denoted by
where a > 0, a ≠ 1, and x is any real number.

4. Properties of Exponents

5. Exponential Function The exponential function
is different from the other functions we have studied so far.
The variable x is an exponent.
A distinguishing characteristic of an exponential function is its rapid increase as x increases (for a > 1).

6. Characteristics of f (x) = ax

7. Characteristics of f (x) = a -x

8. Transformations of f (x) = ax
The graph of g(x) = a(x ± h) is a ______ shift of f.
The graph h(x) = ax ± k is a ________ shift of f.
(Note new horizontal asymptote.)
The graph k(x) = –ax is a reflection of f ______.
The graph j(x) = a(–x) is a reflection of f ______.

9. The Natural Base, e
The natural base (or Euler’s Number) is a constant that occurs frequently in nature and in science.
e ≈
Like π, e is irrational. It continues forever in a non-repeating manner.
The natural base function is f (x) = ex.
The properties and characteristics of exponential functions hold for ex.

10. Graph of ex
Complete the table and graph the function. x ex -2 -1 1 2

11. Applications
Exponential functions are used to model rapid growth or decay.
Examples include:
Compound interest
Population growth
Radioactive decay

12. Compound Interest

13. Example 1
A total of $12,000 is invested at an annual rate of 3%. Find the balance after 4 years if the interest is compounded
Quarterly
Continuously

14. Example 2
Let y represent a mass of radioactive strontium (90Sr), in grams, whose half-life is 28 years. The quantity of strontium present after t years is
What is the initial mass (when t = 0)?
How much of the initial mass is present after 80 years?

15. Example 3
The approximate number of fruit flies in an experimental population after t hours is given by where t ≥ 0.
Find the initial number of fruit flies in the population.
How large is the population of fruit flies after 72 hours?

16. Homework 3.1
Worksheet 3.1