1. Exponential Functions

2. Exponential Functions
y = x2
y = 2x
The function f(x) = bx is an exponential function with base b, where b is a positive real number other than 1 and x is any real number.
An asymptote is a line that a graph approaches (but does not reach) as its x- or y-values become very large or very small.

3. Exponential Functions
Graph y1 = 2x and y2 =
When b > 1, the function f(x) = bx represents exponential growth.
When 0 < b < 1, the function f(x) = bx represents exponential decay.

4. Example 1
Graph f(x) = 2x along with each function below. Tell whether each function represents exponential growth or exponential decay. Then give the y-intercept.
y = 4(2x)
exponential growth, since the base, 2, is > 1
y-intercept is 4 because the graph of f(x) = 2x, which has a y-intercept of 1, is stretched by a factor of 4
exponential decay, since the base, ½, is < 1
y-intercept is 6 because the graph of f(x) = 2x, which has a y-intercept of 1, is stretched by a factor of 6

5. Compound Interest
The total amount of an investment, A, earning compound interest is
where P is the principal,
r is the annual interest rate,
n is the number of times interest is compounded per year,
and t is the time in years.

6. Example 1
Find the final amount of a $500 investment after 8 years at 7% interest compounded annually, quarterly, and monthly.
compounded annually:
= $859.09
compounded quarterly:
= $871.11
compounded monthly:
= $873.91

7. Practice
Find the final amount of a $2200 investment at 9% interest compounded monthly for 3 years.

8. Characteristics of Exponential Functions
The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers.
The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1.
If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing function.
If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing function.
f (x) = bx is a one-to-one function and has an inverse that is a function.
The graph of f (x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote.
f (x) = bx
b > 1
f (x) = bx
0 < b < 1

9. Transformations Involving Exponential Functions
Shifts the graph of f (x) = bx upward c units if c > 0.
Shifts the graph of f (x) = bx downward c units if c < 0.
g(x) = -bx + c
Vertical translation
Reflects the graph of f (x) = bx about the x-axis.
Reflects the graph of f (x) = bx about the y-axis.
g(x) = -bx
g(x) = b-x
Reflecting
Multiplying y-coordintates of f (x) = bx by c,
Stretches the graph of f (x) = bx if c > 1.
Shrinks the graph of f (x) = bx if 0 < c < 1.
g(x) = c bx
Vertical stretching or shrinking
Shifts the graph of f (x) = bx to the left c units if c > 0.
Shifts the graph of f (x) = bx to the right c units if c < 0.
g(x) = bx+c
Horizontal translation
Description
Equation
Transformation

10. Example
Use the graph of f (x) = 3x to obtain the graph of g(x) = 3 x+1.
Solution Examine the table below. Note that the function g(x) = 3x+1 has the general form g(x) = bx+c, where c = 1. Because c > 0, we graph g(x) = 3 x+1 by shifting the graph of f (x) = 3x one unit to the left. We construct a table showing some of the coordinates for f and g to build their graphs.
f (x) = 3x
g(x) = 3x+1
(0, 1)
(-1, 1) 1 2 3 4 5 6 -5 -4 -3 -2 -1

11. Problems Sketch a graph using transformation of the following: 1. 2.3.
Recall the order of shifting: horizontal, reflection (horz., vert.), vertical.

12. The Natural Base e
An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to More accurately,
The number e is called the natural base. The function f (x) = ex is called the natural exponential function.
f (x) = ex
f (x) = 2x
f (x) = 3x
(0, 1)
(1, 2) 1 2 3 4
(1, e)
(1, 3) -1

13. Formulas for Compound Interest
For continuous compounding:
A = Pert

14. Logarithmic Functions & Their Graphs

15. Is this function one to one?
Must pass the horizontal line test.
f(x) = 3x
Is this function one to one?
Yes
Does it have an inverse?
Yes

16. Logarithmic Function of base “a”
Definition:
Logarithmic function of base “a” -
For x > 0, a > 0, and a  1,
y = logax if and only if x = ay
Read as “log base a of x”
f(x) = logax is called the logarithmic function of base a.

17. The most important thing to remember about logarithms is…

18. a logarithm is an exponent.

19. Write the logarithmic equation in exponential form
34 = 81
163/4 = 8
Write the exponential equation in logarithmic form
82 = 64
4-3 = 1/64
log 8 64 = 2
log4 (1/64) = -3

20. Evaluating Logs 2y = 32 2y = 25 y = 5 Think: y = log232 f(x) = log232
Step 1- rewrite it as an
exponential equation.
f(x) = log42
4y = 2
22y = 21
y = 1/2
2y = 32
f(x) = log10(1/100)
Step 2- make the bases the same.
10y = 1/100
10y = 10-2
y = -2
2y = 25
f(x) = log31
Therefore,
y = 5
3y = 1
y = 0

21. Evaluating Logs on a Calculator
You can only use a calculator when the
base is 10
Find the log key on your calculator.

22. Why? log 10 = 1 log 1/3 = -.4771 log 2.5 = .3979 log -2 = ERROR!!!
Evaluate the following using that log key.
log 10 = 1
log 1/3 =
log 2.5 = .3979
log -2 = ERROR!!!
Why?

23. Properties of Logarithms
loga1 = 0 because a0 = 1
logaa = 1 because a1 = a
logaax = x and alogax = x
If logax = logay, then x = y

24. Simplify using the properties of logs
Rewrite as an exponent
4y = 1
Therefore, y = 0
log41=
log77 = 1
Rewrite as an exponent
7y = 7
Therefore, y = 1
6log620 = 20

25. Use the properties of logs to solve these equations.
log3x = log312
x = 12
log3(2x + 1) = log3x
2x + 1 = x
x = -1
log4(x2 - 6) = log4 10
x2 - 6 = 10
x2 = 16
x = 4

26. Review:
How do you find the inverse of a function?
Application of what you know…
What is the inverse of f(x) = 3x?
y = 3x
x = 3y
y = log3x
f-1(x) = log3x
Rewrite the exponential as a logarithm…

27. Find the inverse of the following exponential functions…
f(x) = 2x f-1(x) = log2x
f(x) = 2x f-1(x) = log2x - 1
f(x) = 3x f-1(x) = log3(x + 1)

28. Find the inverse of the following logarithmic functions…
f(x) = log4x f-1(x) = 4x
f(x) = log2(x - 3) f-1(x) = 2x + 3
f(x) = log3x – f-1(x) = 3x+6

29. Graphs of Logarithmic Functions
Graph g(x) = log3x
It is the inverse of y = 3x
Therefore, the table of values for g(x) will be the reverse of the table of values for y = 3x.
y = 3x x y -1
1/3 1 3 2 9
y= log3x x y
1/3 -1 1 3 9 2 
Domain? (0,)  
Range? (-,) 
Asymptotes? x = 0

30. Graphs of Logarithmic Functions
g(x) = log4(x – 3)
What is the inverse exponential function?
y= 4x + 3
Show your tables of values.   
y= 4x + 3 x y -1
3.25 4 1 7 2 19
y= log4(x – 3) x y
3.25 -1 4 7 1 19 2
Domain? (3,)
Range? (-,)
Asymptotes? x = 3

31. Graphs of Logarithmic Functions
g(x) = log5(x – 1) + 4
What is the inverse exponential function?
y= 5x-4 + 1 
Show your tables of values.  
y= 5x-4 + 1 x y 3
1.2 4 2 5 6 26
y= log5(x – 1) + 4 x y
1.2 3 2 4 6 5 26
Domain? (1,)
Range? (-,)
Asymptotes? x = 1

32. Natural Logarithmic Functions
The function defined by
f(x) = logex = ln x, x > 0
is called the natural logarithmic function.

33. Evaluating Natural Logs on a Calculator
Find the ln key on your calculator.

34. Why? ln 2 = .6931 ln 7/8 = -.1335 ln 10.3 = 2.3321 ln -1 = ERROR!!!
Evaluate the following using that ln key.
ln =
ln 7/ =
ln =
ln = ERROR!!!
Why?

35. Properties of Natural Logarithms
ln1 = 0 because e0 = 1
Ln e = 1 because e1 = e
ln ex = x and eln x = x
If ln x = ln y, then x = y

36. Use properties of Natural Logs to simplify each expression
Rewrite as an exponent
ey = 1/e
ey = e-1
Therefore, y = -1
ln 1/e= -1
2 ln e = 2
Rewrite as an exponent
ln e = y/2
e y/2 = e1
Therefore, y/2 = 1 and y = 2. 5
eln 5=

37. Graphs of Natural Log Functions
g(x) = ln(x + 2)
Show your table of values.
y= ln(x + 2) x y -2
error -1
.693 1
1.099 2
1.386  
Domain? (-2,)  
Range? (-,)
Asymptotes? x = -2

38. Graphs of Natural Log Functions
g(x) = ln(2 - x)
Show your table of values.
y= ln(2 - x) x y 2
error 1
.693 -1
1.099 -2
1.386  
Domain? (-2,)  
Range? (-,)
Asymptotes? x = -2