Lecture Notes, Detailed Comments, and Additional Explorations
279
Objectives
Objectives 11.3 LECTURE NOTES
SECTION
Objectives
1. Know the meaning of logarithm and logarithmic function.
SECTION
11.3
LECTURE
1. Know
the 1
meaning
of logarithmNOTES
and logarithmic function.
Math098 Lecture 1.3 1. Know the meaning of logarithm and logarithmic function.
2. Find logarithms and evaluate logarithmic functions.
Objectives
2. Find logarithms and evaluate logarithmic functions.
2. Find logarithms and evaluate logarithmic functions.
3. Know properties of logarithmic functions.
3. Know
properties
offunctions.
logarithmic functions.
logarithmic
1. 4.
Know
meaningand
of logarithm
and logarithmic
function.
2. Graph
Findthe
logarithms
evaluate logarithmic
functions.
4. Graph logarithmic functions.
4. Graph
logarithmic
functions.
logarithms
to of
model
authentic
situations.
2. 5.
Find
logarithms
and
evaluate
logarithmic
functions.
3. Use
Know
properties
logarithmic
functions.
5. Use logarithms to model authentic situations.
5. Use logarithms to model authentic situations.
3. Know
properties
of logarithmic
4. Graph
logarithmic
functions. functions.
Main point: Know the meaning of logarithms and logarithmic functions.
4. Main
5. Use
logarithms
to
model
authentic
situations.
Graph
logarithmic
functions.
point: Know the meaning of logarithms and logarithmic functions.
Main
point:
Know
the meaning
logarithms pairs
and logarithmic
OBJECTIVE
1 Create
tables ofofinput-output
for f (x) = functions.
2x and f 1 (see Tables 2.7 and 2.8).
5. Use logarithms to model authentic situations.
OBJECTIVE 1 Create tables of input-output pairs for f (x) = 2xx and f 11 (see Tables 2.7 and 2.8).
OBJECTIVE
1 Create
tables ofofinput-output
pairs
for f (x) =
2 and f (see Tables 2.7 and 2.8).
Main point: Know
the meaning
logarithms and
logarithmic
functions.
Table 2.7: Values of f
Table 2.8: Values of f 1
1
Table
2.7:
Values
of
finput-outputand
2.8: Tables
Values 2.7
of fand
OBJECTIVE
1 the
Create
tablesof
oflogarithms
pairs
for f (x) =functions.
2x andTable
f 1 (see
2.8).
Main point: Know
meaning
logarithmic
1
Table
of f
of f 1
x 2.7: Values
f (x)
xTable 2.8: Values
f (x)
0
0
f2of
(x)
f 10(x)
x 2 x
1
OBJECTIVE 1x0 xCreate tables
input-output
pairs
for
f
(x)
=
2
and
(see Tables
12.71and 2.8).
1 0
1 x0f
f
(x)
1 0 2.7: Values2of2 f
2Table
2 2.8: Valuesfof1f(x)
0
Table
02 1
222021
22 2201
2 10
13x
2231 2
23x2212
f (x)
f 13(x)21 1
Table2 2.7: Values of42f203
Table
2.8:
Values
of f
2
2403
2222
2422203
4 032
134
3
2314
22x
43
x 14
f (x)2422
f 11(x)
24
1 342
1 42
2
4
2
2
2
• So, f (2
0 ) = 3 and f (220) 3= 4.
23 0
0
24 ) = 4.
2 1
3
1
• So, f 11(233 ) = 3 and f 12(2
2
1
4 exponents of powers of 2.
• Conclude 3that
outputs of f 1 are
24 2
4
• So, f 1 (2
)4 = 3 and f 1 (22422) = 4.
2
2
2
1
Conclude
that
outputs
of
1 f 3 are exponents of powers of 2.1
3
• •Explain
that
we
write
f
(8)
as
log
(8)
and
that
we
write
f
(x)
=
log
(x).
2
3that
214 are exponents
22
3
•• Conclude
of powers of 2.
3 outputs of f
f 1 (2
) =we3 write
and ff 1 (2
1 4 ) = 4.
4
• So,
Explain
(x).
4that
2(8) as log (8) and that we write f 1 (x) =2log
4
3.
of logarithmic
Objectives
1. Know
Knowproperties
the meaning
of logarithmfunctions.
and logarithmic function.
2
2
1
•• Explain
thatthat
weoutputs
write fof1f(8)1 as
that
we write
2 (8) andof
Conclude
arelog
exponents
powers
of 2.f (x) = log2 (x).
Definition
Logarithm
4
• So, f 1 (23 ) = 3 and f 1 (2
) = 4.
•Definition
Explain that weLogarithm
write f 1 (8) as log2 (8) and that we write f 1 (x) = log2 (x).
For b > 0, b 6= 1, and a > 0, the1 logarithm logb (a) is the number k such that bk = a.
Logarithm
•Definition
Conclude that outputs
of f are exponents of powers of 2.
For b > 0, b 6= 1, and a > 0, the logarithm logb (a) is the number k such that bk = a.
1
k
In bshort,
ablogarithm
is
exponent.
•For
Explain
write
f 0,
(8)
log2 (8) and
wethe
write
f 1k(x)
= log
Definition
Logarithm
> 0,that
6=we
1, and
a an
>
the as
logarithm
logbthat
(a) is
number
such
that2 b(x).
= a.
In short, a logarithm is an exponent.
OBJECTIVE
Findathe
logarithm.
For
b > 0, b 6=21, and
> 0,
the logarithm logb (a) is the number k such that bk = a.
In short, a logarithm is an exponent.
OBJECTIVE
2 Find the logarithm.
Definition
1. log2 (32) Logarithm 2. log5 (25)
3. log3 (81)
4. log4 (64)
In
short,
a
logarithm
an logarithm.
exponent.
OBJECTIVE 2 Findisthe
2. logarithm
log5 (25) log (a) is the
3. log
3 (81) k such that bk 4.
For b1.>log
0, 2b(32)
6= 1, and a > 0, the
number
= log
a. 4 (64)
b
OBJECTIVE
2 Find the2.logarithm.
1. log2 (32)
log5 (25)
3. log3 (81)
4. log4 (64)
Definition
Common logarithm
In short,
logarithm is an exponent.
log2a(32)
log5 (25)
3. log3 (81)
4. log4 (64)
1.Definition
Common2.logarithm
A common logarithm is a logarithm with base 10. We write log(a) to represent log10 (a).
Definition
Common
logarithm
OBJECTIVE
2 Find
the logarithm.
A common logarithm is a logarithm with base 10. We write log(a) to represent log10 (a).
Definition
Common
logarithm with base 10. We write log(a) to represent log (a).
common
logarithm
is a2.logarithm
1.Alog
log5 (25)
3. log3 (81)
4.10 log4 (64)
2 (32)
A common logarithm is a logarithm with base 10. We write log(a) to represent log10 (a).
Copyright c 2015 Pearson Education, Inc.
Definition
Common logarithm
Copyright c 2015 Pearson Education, Inc.
A common logarithm is a logarithm
with base
10. We
writeEducation,
log(a) to Inc.
represent log10 (a).
c 2015
Copyright
Pearson
Copyright c 2015 Pearson Education, Inc.
280280
280
280
280
CHAPTER
Lecture
Notes,
Detailed
Comments,
and Additional
Explorations
CHAPTER 22 Lecture
Notes,
Detailed
Comments,
and Additional
Explorations
CHAPTER
2
Lecture
Notes,
Detailed
Comments,
and
Additional
Explorations
CHAPTER
Notes,
Detailed
Comments,
and
Additional
Explorations
CHAPTER 2 2 Lecture
Lecture
Notes,
Detailed
Comments,
and
Additional
Explorations
Find
thethelogarithm.
Find
logarithm.
Find
thethe
logarithm.
Find
logarithm.
Findthe
logarithm.
✓ ◆✓ ◆
✓◆1 ◆◆ 1
log(10,
000)
7.7. log
✓
5. 5.log(10,
000)
log
(6)
6 (6)
6
9. log
9.41✓log
5. 5.5.log(10,
000)
7.
log
(6)
log(10,
000)
7.
log
(6)
114 16
66 (6)
log(10, 000)
7. log
9. 9.
loglog
9.
log4 16
6
4
4
16
p
16 p Explorations
274
CHAPTER 22 Lecture
Notes,
and
Additional
280
CHAPTER
Lecture
Notes,Detailed
DetailedComments,
Comments,
and
Additional
Explorations
16
6.
log(0.0001)
8.
log
(1)
10.
log
3
p
p
8
3
6. 6.log(0.0001)
8.
log
(1)
10.
log
3
p
8
3
log(0.0001)
8.
10.10.
loglog
6. 6.log(0.0001)
8.8.log
log
(1)
10.
log3 3 33
88(1)
3
log(0.0001)
log
8 (1)
3
SECTION
11.2 LECTURE NOTES
Find the logarithm.
OBJECTIVE 3
OBJECTIVE
OBJECTIVE
OBJECTIVE
OBJECTIVE333
✓ ◆
1
7. log6 (6)
9. log4
16
Properties of Logarithms
of Logarithms
1. Know the meaning of inverse of a Properties
function,
invertible
function, and one-to-onepfunction.
Properties
of Logarithms
Properties
of
of Logarithms
Logarithms
6. For
log(0.0001)
8. logProperties
10. log3
3
b > 0 and b 6= 1,
8 (1)
For2.b Know
> 0 and
b
=
6
1,
the
For
b>
6=6=1,1,1, property of inverse functions.
b>
0and
and
breflection
ForFor
00 and
bb6=
• logb (b) = 1
logb (b)the
= inverse
1
3.• Graph
of a function.
=111
=
• ••log
loglog
(b)
=
b (b)
bb(b)
OBJECTIVE
3
log
(1)
=
0
logbb(1)
=0
4.••Find
an
equation
of the inverse of a model.
log
(1)
•
log
(1)
=
b
• logbb (1) ==000
of Logarithms
5. Find an equation of the inverse of Properties
a function that
is not a model.
Definition
function
For
b > 0 and b 6= Logarithmic
1,Logarithmicfunction
Definition
Definition
Logarithmic
function
Definition
Logarithmic function
Definition
log
1 Logarithmic
b (b) =
A•logarithm
logarithm
function,
baseb,b,isfunction
isa inverse
afunction
function
into
form
f (x)
= log
where
0 and
Main
point:
Know
the meaning
of
ofthat
athat
function.
b (x),
A
function,
base
cancan
bebe
putput
into
thethe
form
f (x)
= log
where
b > b0>
and
b (x),
A
logarithm
function,base
baseb,b,isisaafunction
functionthat
thatcan
can be
be put
put into
into the
b 6=
6=
1.1. (1) function,
A blogarithm
the form
form ff(x)
(x)==log
logbb(x),
(x),where
whereb b>>0 0and
and
• log
=0
b 6= 1. b
A
logarithm
function,
base
b,
is
a
function
that
can
be
put
into
the
form
f
(x)
=
log
where
b > 0 and
OBJECTIVE
1
An
employee
is
paid
$10
per
hour.
Let
I
=
f
(t)
be
the
employee’s
income
(in
dollars)
from
b (x),
b 6= 1.
working
t
hours.
b 6= 1.
Discusswhy
whylog
log3 (0)
andlog
log
undefined.
•• Discuss
areare
undefined.
3 (0)and
3 (3 ( 9)9)
••The
function
f log
sends
valueslog
of t( (input
values)
to values of I (output values). For example, f sends 8 to 80
Definition
Discuss
whyLogarithmic
9) are
undefined.
3 (0) andfunction
• Discuss
why of
logwork,
(0)
and
log3 (3 9)earns
are undefined.
3
(for
8
hours
the
employee
$80).
Thedomain
domainofofa alogarithmic
logarithmicfunction
function
positive
numbers.
•• The
loglogisb is
thethe
setset
of of
allall
positive
realreal
numbers.
• Discuss
why log
(0) and logfunction
( 9) areb undefined.
• The domain
of a3 logarithmic
is
the
set
of
all
positive
real
numbers.
A logarithm
function,
base b, is a3functionlog
that
can
be
put
into
the
form
f
(x)
= log (x), where b > 0 and
• The domain of a logarithmic function logbbis the set of all positive real numbers.b
b 6=
1.
•Refer
The to
inverse
of2.7
f isand
a function
thatnotes
sends
ofthe
fthe
to
inputs
ofproperty.
fproperty.
. For example, the inverse of f sends 80
2.8
tooutputs
explain
following
Refer
toTables
Tables
2.7
and
2.8ininthese
these
notes
to
explain
following
• The
of2.7
a the
logarithmic
function
log
of all positive
b is the
to 8domain
(to
earn $80,
must
workto
8explain
hours).
Refer
to Tables
andemployee
2.8 in these
notes
theset
following
property.real numbers.
Refer to Tables 2.7 and 2.8 in these notes to explain the following property.
Logarithmic
Functions
Are
Inverses
of Each
Other
Logarithmicand
andExponential
Exponential
Functions
Are
Inverses
of Each
Other
Refer
to Tables
2.7
and
2.8
in
these
notes
to
explain
the
following
property.
Show
input-output
diagrams
similar
to
Figs.
12
and
13
in
the
textbook
(page
667)
for f and the inverse of
• •Discuss
why log
(0)
and
log
(
9)
are
undefined.
Logarithmic
3
3and Exponential Functions Are Inverses of Each Other
f•. For an exponential
Logarithmic andf (x)
Exponential
Functions
Are Inverses of Each Other
1 = log (x).
bxb, xf, f1 (x)
• For an exponentialfunction
function f (x)==
(x) = log
b b (x).
• The• domain
of Logarithmic
a logarithmic
function
log
set of
numbers.
For an exponential
function
f (x)
= bbxis
, fthe1 (x)
= all
logpositive
and
Exponential
Arereal
Inverses
of Each Other
b (x).
1 Functions
1 a=ifbx
x = b. If the inverse of a function f is also a
•
For
a
logarithmic
function
g(x)
=
log
g
.
x(x),
1 gb(x)
The
inverse
of
a
function
f
is
a
relation
that
sends
to
f
(a)
b
•
For
a
logarithmic
function
g(x)
=
log
(x),
(x)
=
b
.
• For an exponential function f (x) = b ,bf (x) = logb (x).
1 the inverse
say
f 2.7
is invertible
use
“f =1 ”to
asexplain
name
for
of f . We say f 1 is the inverse function For
a logarithmic
function
g(x)
log
g the
(x) = bx . property.
ba(x),
Refer•towe
Tables
and 2.8
inand
these
notes
function,
x
11 following
x
For aan
exponential
function
=b b(x),
, fg (x)
• For
logarithmic
function
g(x)f (x)
= log
(x) =
= blog. b (x).
.•
of f Find
Findthe
theinverse
inverseofofthe
thefunction.
function.
Find
thea inverse
of the function.
• For
function
g 1h(x)
(x)
=log
bx . (x) of Each
and g(x)
Exponential
Functions
Are
Inverses
Other
x= logb (x),
11.
(x)
==logarithmic
55xx Logarithmic
12.
13.
=
14. 14.
k(x)
= log(x)
7
Find
of the function.
11. ffthe
(x)inverse
12.g(x)
g(x)=
=1010x of an Inverse
13. h(x)
= log
k(x)
= log(x)
Property
Function
7 (x)
x
x
11. f (x) = 5
12. g(x) = 10 x
13.
h(x)
=
log
(x)
14.
k(x)
=
log(x)
7
x
• For
an
exponential
function
f (x) = xb , f 1 (x) = logb (x).
x
1
Let
f
(x)
=
4
.
Find
the
inverse
of
the
function.
x
anfinvertible
f12.
, theg(x)
following
f (a)
= b and f14.(b)
= a.= log(x)
11.ForfLet
(x)
= 5= 4 function
= 10 statements are
13.equivalent:
h(x) = log
k(x)
7 (x)
(x)
.
x
1
x
Let
f
(x)
=
4
.
• For a logarithmic
function g(x) =1 logbx(x), g (x) = b .
xx
15.f (x)
Find=
f (4).
16.
Find
f =(4).
17. Find
fh(x)
(3). = log7 (x)18. Find 14.
f 1 (16).
11.Let
5(4).
12.
g(x)
= log(x) 1
f
(x)
=
4
.
15.
Find
f
16. Find
f 110
(4).
17.13.
Find
f (3).
18. Find fk(x)
(16).
15.
Find
f (4).
16. Find
f 1f(4).
18. Find f 1 (16).
1. Let
be
an invertible
function
where
(5) = 7. Find17.
f 1Find
(7). f (3).
Find
thefinverse
of the function.
LetFind
f (x)
= 4x .4 Because
15.
f (4).
16.a logarithmic
Find f 1 (4).
17.inverse
Find fof
(3).
Find fwe 1can
(16).
OBJECTIVE
function is the
an exponential 18.
function,
use the
x
x
OBJECTIVE
4 Because
a logarithmic
functioninverse
is the
inverse
an
exponential
wefunction.
can use the
11.
f values
(x) graphing
= of
5 an invertible
g(x)
=are
10for
13.
h(x)
=
logof7to(x)
14. k(x)
= log(x)
four-step
method
of12.
Section
11.2
graphing
functions
help
us graph
afunction,
logarithmic
Some
function
f
shown
in
the
following
table.
OBJECTIVE
4 BecauseSection
a logarithmic
the inverse
of antoexponential
function,
we can use the
1 function
four-step
graphingisinverse
help us graph
a logarithmic
15.
Find fgraphing
Find11.2
f for
(4).
17. functions
Find f (3).
18.
Find f 1function.
(16). Graph
by(4).
hand. method of16.
four-step
graphing
method
of
Section
11.2
for
graphing
inverse
functions
to
help
us
graph
a
logarithmic
function.
x 4 Because a logarithmic function is the inverse of an exponential function, we can
OBJECTIVE
use the
LetGraph
f (x) by
= 4hand.
.
x
f (x)
2. Find f (3).
Graph
by hand.
four-step
graphing
method
of
Section
11.2
for
graphing
inverse
functions
to
help
us
graph
a
logarithmic
function.
19. y = log3 (x) 2
20. y = log 141 (x)
3
3.20.Find
f log(3).
19.
y=
log
y=
(x)
1
1
Graph
by
hand.
3 (x) 3
6
15.
Find
f
(4).
16.
Find
f
(4).
17.
Find
finverse
(3).114(x)
18. Find
f function,
(16). we can use the
OBJECTIVE
4
Because
a
logarithmic
function
is
the
of an exponential
1
19. y = log3 (x)
20.
y
=
log
4.
Create
a
table
of
values
of
f
.
4
4
four-step graphing method
of Section12
11.2 for graphing inverse functions to help us graph a logarithmic function.
c 2015 Pearson
Copyright
Inc.
5
24
19. y = log3 (x)
20. yEducation,
= log 14 (x)
Graph by hand. 6
c 2015 Pearson Education,
Copyright
Inc.
48
OBJECTIVE 4 Because a logarithmic
the inverse
of an exponential
function, we can use the
c 2015isPearson
Copyrightfunction
Education,
Inc.
four-step graphing method of Section 11.2 for graphing inverse functions to help us graph a logarithmic function.
5.Objectives
log(10, 000)
11. f (x) = 5x
12. g(x) = 10x
13. h(x) = log7 (x)
14. k(x) = log(x)
16. Find f
17. Find f (3).
18. Find f
Let f (x) = 4x .
15. Find f (4).
1
(4).
1
(16).
OBJECTIVE
4 Because a logarithmic function is the inverse of an exponential function, we can use the
four-step
graphing method of Section 11.2 for graphing inverse functions to help us graph a logarithmic function.
Graph
by hand.
Sketch the graph of y = log3(x). 19. y = log3 (x)
20. y = log 14 (x)
The inverse of f(x) = 3x is f –1(x) = log3(x). So, we can apply the four-­‐step method to graph the inverse of f: Step 1: Sketch the graph of f. Copyright c 2015 Pearson Education, Inc.
Step 2: Choose several points on the graph of f: A sketch of f with these points is shown on the next slide. Step 3: For each point (a, b) chosen in step 2, plot point (b, a). Step 4: A sketch the curve that contains the points plotted in step 3 is shown on the next slide. The red curve is the graph of y = log3(x).