1. Exponential and Logarithmic Functions
Exponential Functions & Their Graphs
Logarithmic Functions & Their Graphs
Properties of Logarithms
Exponential and Logarithmic Equations
Exponential and Logarithmic Models

2. Exponential Functions
Definition of Exponential Functions
The exponential function f with base a is denoted by
where a > 0, a ≠ 1, and x is any real number.
Example 1 Evaluating Exponential Expressions
Use a calculator to evaluate each expression.
a b.
Rounded to nearest ten thousandth
a b.

3. Graphs of Exponential Functions
The graphs of all exponential functions have similar characteristics.
Example 2 Graphs of y = ax
Create a table x -3 -2 -1 1 2 2x 4x
Graph of y = 2x
Graph of y = 4x
Combined Graphs

4. Graphs of Exponential Functions
The graphs of all exponential functions have similar characteristics.
Example 2 Graphs of y = a- x
Create a table x -3 -2 -1 1 2
2-x
4-x
Graph of y = 2-x
Graph of y = 4-x
Combined Graphs

5. Comparing Exponential Functions
Basic Characteristics of Exponential Functions
Function
Y = ax, a >1
Y = a-x, a >1
Domain:
(- ∞, ∞)
Range:
(0, ∞)
y-Intercept:
(0, 1)
(0,1)
Increasing/Decreasing
Increasing
Decreasing
Horizontal Asymptote
y = 0
Continuity:
Continuous
Graph of y = ax
Graph of y = a-x

6. Shifting Exponential Functions
Each of the following graphs represents a transformation of the graph of f(x) = 3x
Because g(x) = 3x+1 = f(x+1) , the graph of g can be obtained by shifting the graph of f one unit to the left.
Because g(x) = 3x - 2 = f(x) - 2 , the graph of g can be obtained by shifting the graph of f one unit to the left.

7. Shifting Exponential Functions
Each of the following graphs represents a transformation of the graph of f(x) = 3x
Because g(x) = 3x - 2 = f(x) - 2 , the graph of g can be obtained by shifting the graph of f one unit to the left.
Because g(x) = - 3x = - f(x) , the graph of g can be obtained by shifting the graph of f one unit to the left.

8. The Natural Base e
In many applications the most convenient choice for a base is the irrational number e. e ≈
This number is known as the natural base. The function f(x) = ex is called the natural exponential function.
Example 3 Evaluating the Natural Exponential Function
Graph of the Natural Base
Use a calculator to evaluate each expression.
a b.
c d.
Rounded to nearest ten thousandth
a b.
c d.

9. Graphing Exponential Functions
Example 4 Graphing natural Exponential Functions
Graph each natural exponential function
a b.
Create a Table X -3 -2 -1 1 2 3
f(x)
g(x)
Graph of f(x)
Graph of g(x)

10. Applications of Exponential Functions
One of the most familiar examples of exponential growth is that of an investment earning continuously compounded interest.
Formulas for Compound Interest
After t years the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas. 1. 2.
Example 5 Compounding n times and Continuously
A total of $12,000 is invested at an interest rate or 9%. Find the balance after 5 years if it is compounded quarterly and continuously.
Quarterly:
Formula:
Substitution:
Solution:
Continuously:
Formula:
Substitution:
Solution:
Note that continuous compounding yields $93.64 more than quarterly.

11. Applications of Exponential Functions
Example 6 Radioactive Decay
In 1986, a nuclear reactor accident occurred in Chernobyl in what was then the Soviet Union. The explosion spread radioactive chemicals over hundreds of square miles, and the government evacuated the city and surrounding areas. To see why the city is now uninhabited, consider the following model.
This model represents the amount of plutonium that remains of the initial 10 pounds after t years.
How many grams of plutonium will remain after 100 years?
How many grams of plutonium will remain after 10,000 years?
From the graph you can see that Plutonium has a half life of 24,360 years.

12. Logarithmic Functions
Previously we have studied functions and their inverses. During this study we realized that if a function has the property were no horizontal line intersect the graph more than once the function has an inverse.
Looking at the graph of f(x) = ax we notice that f(x) has an inverse.
Definition of Logarithmic Function
For x>0 and a≠1,
if and only if
The function given by is called the logarithmic function with base a.
When evaluating logarithms, remember that a logarithm is an exponent. This means that logax is the exponent to which a must be raised to obtain x.
For instance, log28=3 because 2 raised to the 3 power is 8.

13. Evaluating Logarithms
Example 7 Evaluating Logarithms
Expression
Value
Justification a. b. c. d. e. f. 5 3 -2 1

14. Evaluating Logarithms
Example 7 Evaluating Logarithms on the Calculator
Use the calculator to evaluate each expression.
Expression
Key Strokes
Display a. b. c.
LOG ENTER 1
2 X LOG ENTER
LOG (-) ENTER
ERROR
Properties of Logarithms
because
then

15. Graphing Logarithmic Functions
Example 8 Graphing a Logarithmic Function
In the same coordinate plane graph the following two functions.
a b. x -2 -1 1 2 3 2x
log10x

16. Natural Logarithmic Function
The Natural Logarithmic Function
The function defined by
is called the natural logarithmic function.
Properties of Natural Logarithms
ln 1 = 0
ln e = 1
ln ex = x
If ln x = ln y, then x = y
Example 9 Using Properties of Natural Logarithms
Example
Solution
Property a. b. c. d.
Property 3
Property 1
Property 2

17. Domains of Logarithmic Functions
Example 10 Finding the Domains of Logarithmic Functions
Find the domain of each function
a. f(x) = ln (x – 2)
b. g(x) = ln (2 – x)
c. Ln x2
Because ln (x – 2) is defined only if x – 2 > 0, it follows that the domain of f is (2, ∞).
Because ln (2 – x) is defined only if 2 – x > 0, it follows that the domain of g is (- ∞, 2).
Because ln x2 is defined only if x2 > , it follows that the domain of h is all real numbers except x = 0

18. Application of Natural Logarithms
Example 11 Human Memory Models
Students participating in a psychological experiment attended several lectures on a subject and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model.
where t is time in months.
a. What was the original average score?
c. What was the average score after six months?
b. What was the average score after two months?

19. Change of Base Change of Base Formula
Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1.
Then
Example 12 Changing Bases Using Common Logarithms
Change each logarithmic function to base 10 and evaluate to nearest ten thousandth.
Example
Change of Base
Evaluate a. b.

20. Properties of Logarithms
Let a be a positive number such that a ≠ 1, and let n be a real number. If u and v are positive real numbers, the following properties are true. 1. 2. 3. 1. 2. 3.
Example 13 Using the Properties of Logarithms
Write the logarithm in terms of ln 2 and ln 3 a. b.

21. Using Properties of Logarithms
Example 14 Using Properties of Logarithms
Use the properties of logarithms to verify that
Change one side to match the other.
Original Statement
Law of Negative Exponents
ln un = n ln u
Simplify
Example 15 Rewrite Each Logarithm In Expanded Form
Original Statement
Product Rule
Power Rule
Original Statement
Quotient Rule
Power Rule

22. Using Properties of Logarithms
Example 16 Rewrite each Logarithmic Expression in Condensed Form
Original Statement
Power Rule Of Logarithms
Product Rule of Logarithms
Example 17 Rewrite Each Logarithm In condensed Form
Original Statement
Product Rule
Power Rule

23. Exponential and Logarithmic Equations
Solving Exponential Equations
Solving Logarithmic Equations
Isolate the exponential expression
Take the logarithm of both sides
Solve for the variable
Rewrite the equation in exponential form
Solve for the variable
Example 18 Solving an Exponential Equation
Solve
Original Equation
Take logarithm of both sides
Inverse Property
Evaluate to the thousandths place
Example 19 Solving an Exponential Equation
Solve
Original Equation
Isolate the exponential expression
Take the logarithm of both sides
Inverse Property
Evaluate to the thousandths place

24. Exponential and Logarithmic Equations
Examples 20 & Solving an Logarithmic Equation
Solve
Original Equation
Solve
Isolate the exponential expression
Evaluate to nearest thousandths
Take logarithm of both sides
Inverse Property
Solve
Original Equation
Quadratic Form
Factor
Set factors equal to zero
Isolate exponential expression
Take logarithm of both sides
Evaluate to nearest thousandths

25. Exponential and Logarithmic Equations
Example 20 Solving an Logarithmic Equation
Solve
Original Equation
Isolate the Natural Logarithm
Exponentiate both sides
Inverse Property
Solve
Evaluate to the thousandths place
Solve
Original Equation
Quotient Rule for Logarithms
Exponentiate both sides
Cross Multiply
Subtract ex from both sides
Factor
Divide both sides by 1 - e

26. Applications Example 21 Doubling an Investment
You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double ? How long for it to triple?
Double Investment
Triple Investment
Formula
Formula
Substitute
Substitute
Isolate
Isolate
Take Logarithm
Take Logarithm
Inverse Property
Inverse Property
Solve
Solve
Simplify
Simplify

27. Application Example 22 Consumer Price Index for Sugar
From 1970 to 1973, the consumer Price Index ( CPI ) value y for a fixed amount of sugar for the year t can be modeled by the equation
where t = 10 represent During which year did the price of sugar reach four times its 1970 price of 30.5 on the CPI?
Formula
Substitute
Isolate
Take Logarithm
Inverse Property
Solve
Since t = 0 represents 1970, the price of sugar reached 4 times its1079 price in 1988.