Logarithmic Function a REVIEW powerpoint

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

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

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:

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:

Exponential Equations
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:

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:

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:

• General Form of an Exponential Function:
f (x) = Nx
, N > 0
g(x) = 2x
x
2x
g(3) =
g(2) =
g(1) =
g(0) =
g(–1) =
g(–2) =
8
4
2
1
1
2 1
2

2
2
2
1
2
 1
4


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

g(x) = 2x

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

h(x) = 3x

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

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

The Exponential Function
f (x) = ex

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.

A logarithmic function is a function
defined by y = logb x, if and only if x = by
for all positive real numbers x and b, and
b ≠ 1.
Definition of Logarithmic Function

A logarithmic function is a function
defined by y = logb x, if and only if x = by
for all positive real numbers x and b, and
b ≠ 1.
Definition of Logarithmic Function
“y is equal to the logarithm of x to the base b”

A logarithmic equation in one variable
is an equation involving logarithms of
expression containing the variable.
Solving Logarithmic Equations

A logarithmic equation in one variable
is an equation involving logarithms of
expression containing the variable.
Solving Logarithmic Equations
Exponential Equation Logarithmic Equation
22
= 4 log2 4 = 2
103
= 1000 log10 1000 = 3
25½
= 5 log25 5 = ½
4x
= 16 log4 16 = x

Express each exponential equation in its equivalent
logarithmic equation.
70
= 1

70
= 1 log7 1 = 0

70
= 1 log5 1 = 0
35
= 243

70
= 1 log5 1 = 0
35
= 243 log3 243 = 5

70
= 1 log5 1 = 0
35
= 243 log2 32 = 5
ab
= c

70
= 1 log5 1 = 0
35
= 243 log2 32 = 5
ab
= c loga c = b

70
= 1
35
= 243
ab
= c
16-3/4
= 1/8

70
= 1
35
= 243
ab
= c
16-3/4
= 1/8 log16 1/8 = -¾

Logarithmic Equation Exponential Equation
log5 125 = 3

log5 125 = 3 53
= 125

log5 125 = 3 53
= 125
log135 2460375 = 3

log5 125 = 3 53
= 125
log135 2460375 = 3 1353
= 2460375

log5 125 = 3 53
= 125
log135 2460375 = 3 1353
= 2460375
logr q = s

log5 125 = 3 53
= 125
log135 2460375 = 3 1353
= 2460375
logr q = s rs
= q

log5 125 = 3 53
= 125
log135 2460375 = 3 1353
= 2460375
logr q = s rs
= q
log6 1/1296 = -4

log5 125 = 3 53
= 125
log135 2460375 = 3 1353
= 2460375
logr q = s rs
= q
log6 1/1296 = -4 6-4
= 1/1296

The logarithm of a number N to the
base b is the exponent of the power to
which b is raised to obtain N. In symbols,
logb N = x if and only if bx
= N.
The Logarithm of a Number

Evaluate the following logarithms.
log1/2 64

Solve each logarithmic equation.
logn 1/8 = -3

Written Work 3.2
Answer Reinforcement 3.2.2 (A, even numbers
only) (B, odd numbers only) and (C, 6 - 10 only)
on pages 106 - 107.

Let b be a positive real number and b ≠ 1.
Then,
logb M = logb N
if and only if M = N.
Property of Equality for
Logarithmic Equations

Solve each logarithmic equation.
log5 (x + 1)2
= log5 (x2
– 3)

1st
Law of Logarithms
For any base b,
i. logb b = 1
ii. logb 1 = 0
iii. logb bx
= x (b > 0, b ≠ 1)
Laws of Logarithm

2nd
Law of Logarithms: Logarithm of
Products
logb MN = logb M + logb N
Laws of Logarithm

Express as the sum of simpler logarithmic
expressions.
log3 (6xz)

Express as a single logarithm.
log5 m + log5 12 + log5 n

3rd
Quotients
logb M/N = logb M – logb N
Laws of Logarithm

Express as the sum or difference of simpler
logarithmic expressions.
log7 20/3

Express the difference of logarithms as a single
logarithm.
log a – log (3a – 1)

4th
Power
logb Ma
= alogb M
Laws of Logarithm

Express as a sum or difference of logarithms.
logm (n – 2)½
(4n)

Express each as a single logarithm.

Logarithmic Function a REVIEW powerpoint

More Related Content

Similar to Logarithmic Function a REVIEW powerpoint

More from MarkAnthonyBIsrael

Recently uploaded

Logarithmic Function a REVIEW powerpoint