Exponential_Functions general mathematics

Exponential Functions

Objectives
Objectives
To use the properties of exponents to:
To use the properties of exponents to:
 Simplify exponential expressions.
Simplify exponential expressions.
 Solve exponential equations.
Solve exponential equations.
To sketch graphs of exponential functions.
To sketch graphs of exponential functions.

A polynomial function has the basic form:
A polynomial function has the basic form: f
f (
(x
x) =
) = x
x3
3
An exponential function has the basic form:
An exponential function has the basic form: f
f (
(x
x) = 3
) = 3x
x
An exponential function has the variable in the
An exponential function has the variable in the exponent
exponent,
,
not
not in the
in the base
base.
.
General Form of an Exponential Function:
f
f (
(x
x) =
) = N
Nx
x
, N >
, N > 0
0

Properties of Exponents
X Y
A A
  X Y
A 
XY
A
X Y
A 
 
Y
X
A 
X
Y
A
A

 
X
AB  X X
A B
X
A
B
 

 
 
X
X
A
B
X
A

1
X
A
1
X
A
 X
A
X
Y
A Y X
A  
X
Y
A


2 3
2 2
  5
2 32

2 6
2 2
  4
2
4
1
2

1
16

 
2
3
2  6
2 64

Simplify:
Simplify:

3
2
3

 

 
 
3
3
2
3


3
3
3
2

7
9
3
3
 2
3
2
1
3

1
9

  
1 1
2 2
2 8 
1
2
16 16

27
8

4

 
1
2
2 8
 
Simplify:
Simplify:

Exponential Equations
Solve:
Solve: Solve:
Solve:
5 125
x

3
5 5
x

3
x 
1
2
( 1)
7 7
x 


1
2
1
x  
1
2
x 

8 4
x

 
3 2
2 2
x

3 2
x 
2
3
x 
3 2
2 2
x

8 2
x

 
3 1
2 2
x

3 1
x 
1
3
x 
3 1
2 2
x

Solve:
Solve: Solve:
Solve:

1
3
27
x 
 
1
3
3
3
27
x 
19,683
x 
 
1
3 27
x

 
1
3 27
x


3
x
 
3
x 
3
3 3
x


Not considered an
exponential equation,
because the variable
is now in the base.
Solve:
Solve: Solve:
Solve:

3
4
8
x 
 
4
3 4
3
3
4
8
x 
 
4
3
8
x 
 
4
2
x 
16
x 
Not considered an
exponential equation,
because the variable
is in the base.
Solve:
Solve:

f
f (
(x
x) =
) = N
Nx
x
, N >
, N > 0
0
g
g(
(x
x) = 2
) = 2x
x
x
2
2x
x
g
g(3) =
(3) =
g
g(2) =
(2) =
g
g(1) =
(1) =
g
g(0) =
(0) =
g
g(–1) =
(–1) =
g
g(–2) =
(–2) =
8
4
2
1
1
2 1
2

2
2
2
1
2
 1
4


g
g(
(x
x) = 2
) = 2x
x
x
2
1
0
4
2
1
 
g x
–1
–2
1
2
1
4

g
g(
(x
x) = 2
) = 2x
x

h
h(
(x
x) = 3
) = 3x
x
x
2
1
0
9
3
1
 
h x
–1
–2
1
3
1
9

h
h(
(x
x) = 3
) = 3x
x

Exponential functions with
Exponential functions with
positive bases
positive bases greater
greater than
than
1 have graphs that are
1 have graphs that are
increasing
increasing.
.
The function never crosses
The function never crosses
the
the x
x-axis because there is
-axis because there is
nothing we can plug in for
nothing we can plug in for x
x
that will yield a zero answer.
that will yield a zero answer.
The
The x
x-axis is a left
-axis is a left
horizontal asymptote.
horizontal asymptote.
h
h(
(x
x) = 3
) = 3x
x
(red)
(red)
g
g(
(x
x) = 2
) = 2x
x
(blue)
(blue)

A
A smaller
smaller base means
base means
the graph rises more
the graph rises more
gradually
gradually.
.
A
A larger
larger base means the
base means the
graph rises more
graph rises more quickly
quickly.
.
Exponential functions will
Exponential functions will
not
not have
have negative
negative bases.
bases.
h
h(
(x
x) = 3
) = 3x
x
(red)
(red)
g
g(
(x
x) = 2
) = 2x
x
(blue)
(blue)

The Number
The Number e
e
e 
2.71828169
A base often associated with exponential functions is:

The Number
The Number e
e
Compute:
Compute:  
1
0
lim 1 x
x
x


x
–.1
–.01
–.001
2.868
2.732
2.7196
 
1
1 x
x

 
1
0
lim 1 x
x
x



x
.1
.01
.001
2.5937
2.7048
2.7169
 
1
1 x
x

 
1
0
lim 1 x
x
x



2.71828169


The Number
The Number e
e
Euler’s number
Euler’s number
Leonhard Euler
Leonhard Euler
(pronounced “oiler”)
(pronounced “oiler”)
Swiss mathematician
Swiss mathematician
and physicist
and physicist

The Exponential Function
The Exponential Function
f
f (
(x
x) =
) = e
ex
x

x
2
1
0
4
2
1
 
j x
–1
–2
1
2
1
4
   
1
2
x
j x 

   
1
2
x
j x 
Exponential functions with positive bases less
than 1 have graphs that are decreasing.

Why study exponential functions?
Why study exponential functions?
Exponential functions are used in our real world
Exponential functions are used in our real world
to measure growth, interest, and decay.
to measure growth, interest, and decay.
Growth obeys exponential functions.
Growth obeys exponential functions.
Ex: rumors, human population, bacteria,
Ex: rumors, human population, bacteria,
computer technology, nuclear chain reactions,
computer technology, nuclear chain reactions,
compound interest
compound interest
Decay obeys exponential functions.
Decay obeys exponential functions.
Ex: Carbon-14 dating, half-life, Newton’s Law of
Ex: Carbon-14 dating, half-life, Newton’s Law of
Cooling
Cooling

Exponential_Functions general mathematics

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