abcsdsfsGeneral-Mathematics-Week-7_1.pdf
SyrusLeonardPasawaJu
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Sep 09, 2024
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About This Presentation
General mathematics grade 11
Slide Content
“To educate children, you must
love them and love them all
equally.”
-St. Marcellin Champagnat
Notre Dame of
Kidapawan College
Integrated Basic Education
Department
General Mathematics
Week 7
love them and love them all
equally.”
-St. Marcellin Champagnat
Notre Dame of
Kidapawan College
Integrated Basic Education
Department
General Mathematics
Week 7
Notre Dame of Kidapawan College
Integrated Basic Education Department | 2
Solve for the value of x
in the equation
1
2
??????
=32.
Ans.: X=-5
Solve the inequality
3
2�−4
<9.
Ans.: x<3
Solve the equation
5
2+2�
=
1
5
.
Ans. : x=−
�
�
Integrated Basic Education Department | 2
Solve for the value of x
in the equation
1
2
??????
=32.
Ans.: X=-5
Solve the inequality
3
2�−4
<9.
Ans.: x<3
Solve the equation
5
2+2�
=
1
5
.
Ans. : x=−
�
�
Notre Dame of Kidapawan College
Integrated Basic Education Department | 3
Logarithmic
Functions,
Equations and
Inequalities
Integrated Basic Education Department | 3
Logarithmic
Functions,
Equations and
Inequalities
Notre Dame of Kidapawan College
Integrated Basic Education Department | 4
Represent real-life situations using logarithmic function.
Distinguish logarithmic function, logarithmic equation,
and logarithmic inequality.
Solve logarithmic equations and inequalities.
Learning Competencies
Represent logarithmic function through its: (a) table of
values, (b) graph, and (c) equation.
Find the domain and range of a logarithmic function.
Determine the intercepts, zeroes, and asymptotes of
logarithmic function.
Solve problems involving logarithmic function,
logarithmic equation, and logarithmic inequality.
Integrated Basic Education Department | 4
Represent real-life situations using logarithmic function.
Distinguish logarithmic function, logarithmic equation,
and logarithmic inequality.
Solve logarithmic equations and inequalities.
Learning Competencies
Represent logarithmic function through its: (a) table of
values, (b) graph, and (c) equation.
Find the domain and range of a logarithmic function.
Determine the intercepts, zeroes, and asymptotes of
logarithmic function.
Solve problems involving logarithmic function,
logarithmic equation, and logarithmic inequality.
Notre Dame of Kidapawan College
Integrated Basic Education Department | 5
Thinkofalogarithmofxtothebaseb(denoted
by??????��
��)astheexponentofbthatgives??????.For
example,??????��
381=4because3
4
=81.Some
additionalexamplesaregivenbelow:
(a)??????��
51=0because5
0
=1
(b)??????��
6
1
6
=−1because6
−1
=
1
6
Concept of Logarithm
Integrated Basic Education Department | 5
Thinkofalogarithmofxtothebaseb(denoted
by??????��
��)astheexponentofbthatgives??????.For
example,??????��
381=4because3
4
=81.Some
additionalexamplesaregivenbelow:
(a)??????��
51=0because5
0
=1
(b)??????��
6
1
6
=−1because6
−1
=
1
6
Concept of Logarithm
Notre Dame of Kidapawan College
Integrated Basic Education Department | 6
1.??????��
232
2.??????��
9729
3.??????��
55
Try this!!!
Integrated Basic Education Department | 6
1.??????��
232
2.??????��
9729
3.??????��
55
Try this!!!
Notre Dame of Kidapawan College
Integrated Basic Education Department | 7
Logarithmic Form:??????��
��=�Exponential Form: �
�
=�
??????��
232=5 2
5
=32
??????��
9729=3 9
3
=729
??????��
55=1 5
1
=5
??????��Τ1216=−4 Τ12
−4
=16
??????��
101000=3 10
3
=1000
•In both the logarithmic and exponential forms, b is the base.
•In the exponential form, c is an exponent. But c=??????��
��. This implies
that the logarithm is actually an exponent.
•In the logarithmic form ??????��
��, a cannot be negative. For example,
??????��
2−8is not defined since 2 raised to any exponent will never
result to a negative number.
•The value of ??????��
��can be negative. For example, ??????��
5
1
125
=−3
because 5
−3
=
1
125
.
Integrated Basic Education Department | 7
Logarithmic Form:??????��
��=�Exponential Form: �
�
=�
??????��
232=5 2
5
=32
??????��
9729=3 9
3
=729
??????��
55=1 5
1
=5
??????��Τ1216=−4 Τ12
−4
=16
??????��
101000=3 10
3
=1000
•In both the logarithmic and exponential forms, b is the base.
•In the exponential form, c is an exponent. But c=??????��
��. This implies
that the logarithm is actually an exponent.
•In the logarithmic form ??????��
��, a cannot be negative. For example,
??????��
2−8is not defined since 2 raised to any exponent will never
result to a negative number.
•The value of ??????��
��can be negative. For example, ??????��
5
1
125
=−3
because 5
−3
=
1
125
.
Notre Dame of Kidapawan College
Integrated Basic Education Department | 8
CommonLogarithm
Commonlogarithmsarelogarithmswithbase10.Thebase10
isusuallyomittedwhenwritingcommonlogarithms.Thismeansthat
??????���isashortnotationfor??????��
10�.
Example:??????��100=2because10
2
=100
NaturalLogarithm
Logarithmswithbaseearecallednaturallogarithmsandare
denotedby"??????�“.Inotherwords,ln�isanotherwayofwriting
??????��
??????�.
Example:ln�=2because�
2
=�
Integrated Basic Education Department | 8
CommonLogarithm
Commonlogarithmsarelogarithmswithbase10.Thebase10
isusuallyomittedwhenwritingcommonlogarithms.Thismeansthat
??????���isashortnotationfor??????��
10�.
Example:??????��100=2because10
2
=100
NaturalLogarithm
Logarithmswithbaseearecallednaturallogarithmsandare
denotedby"??????�“.Inotherwords,ln�isanotherwayofwriting
??????��
??????�.
Example:ln�=2because�
2
=�
Notre Dame of Kidapawan College
Integrated Basic Education Department | 9
log
��=�if and only if �
�
=�for b>0,
b≠1.
A logarithm is an exponent which bmust
have to produce x.
In either equation,b is called the base and
must be a positive number, not equal to 1.
Conversion Rules
Integrated Basic Education Department | 9
log
��=�if and only if �
�
=�for b>0,
b≠1.
A logarithm is an exponent which bmust
have to produce x.
In either equation,b is called the base and
must be a positive number, not equal to 1.
Conversion Rules
Notre Dame of Kidapawan College
Integrated Basic Education Department | 10
Write 3
2
=�in logarithmic
form.
Example:
3
2
=�means log
3�=2
The base remains
the same.
Exponents are logarithms
Integrated Basic Education Department | 10
Write 3
2
=�in logarithmic
form.
Example:
3
2
=�means log
3�=2
The base remains
the same.
Exponents are logarithms
Notre Dame of Kidapawan College
Integrated Basic Education Department | 11
Write the logarithmic
equation ??????????????????
�??????=�into
exponential form.
Try This!
Integrated Basic Education Department | 11
Write the logarithmic
equation ??????????????????
�??????=�into
exponential form.
Try This!
Notre Dame of Kidapawan College
Integrated Basic Education Department | 12
Logarithmic Equation Logarithmic Inequality Logarithmic Function
DefinitionAn equation involving
logarithms
An inequality involving
logarithms
Function of the form
��=??????��
��, �>0, �≠1.
Example log
�2=4 ln�
2
>(ln�)
2
��=log
3�
Logarithmic Functions, Equations, and
Inequalities
1.) log
�2<4
2.) ��=log
2�
3.) x=log
255
4.) log
42�=log
410
5.) log
7(�+18)≥4
Logarithmic Inequality
Logarithmic Function
Logarithmic Equation
Logarithmic Inequality
Logarithmic Equation
Integrated Basic Education Department | 12
Logarithmic Equation Logarithmic Inequality Logarithmic Function
DefinitionAn equation involving
logarithms
An inequality involving
logarithms
Function of the form
��=??????��
��, �>0, �≠1.
Example log
�2=4 ln�
2
>(ln�)
2
��=log
3�
Logarithmic Functions, Equations, and
Inequalities
1.) log
�2<4
2.) ��=log
2�
3.) x=log
255
4.) log
42�=log
410
5.) log
7(�+18)≥4
Logarithmic Inequality
Logarithmic Function
Logarithmic Equation
Logarithmic Inequality
Logarithmic Equation
Notre Dame of Kidapawan College
Integrated Basic Education Department | 13
??????��
2�=4
2
4
=�
�=16
Solve the following simple
logarithmic equations:
??????��1000=−�
10
−�
=1000
10
−�
=10
3
−�=3
�=−3
Integrated Basic Education Department | 13
??????��
2�=4
2
4
=�
�=16
Solve the following simple
logarithmic equations:
??????��1000=−�
10
−�
=1000
10
−�
=10
3
−�=3
�=−3
Notre Dame of Kidapawan College
Integrated Basic Education Department | 14
Let�and�berealnumberssuchthat�>0,and
�≠1.
1.??????��
�1=0
Example:??????��1=0
2.??????��
��
�
=�
Example:??????��
464=??????��
44
3
=3
3.If�>0,then�
??????�??????��
=�
Example:5
??????�??????52
=2
Basic Properties of Logarithms
Integrated Basic Education Department | 14
Let�and�berealnumberssuchthat�>0,and
�≠1.
1.??????��
�1=0
Example:??????��1=0
2.??????��
��
�
=�
Example:??????��
464=??????��
44
3
=3
3.If�>0,then�
??????�??????��
=�
Example:5
??????�??????52
=2
Basic Properties of Logarithms
Notre Dame of Kidapawan College
Integrated Basic Education Department | 15
Laws of Logarithms
Laws of Logarithms
Let �>0, �≠1and �∈ℝ.
For �>0, �>0, then: Examples:
1. ??????��
���=??????��
��+??????��
��1. ??????��
327⋅81=??????��
327+??????��
381
2. ??????��
�
�
�
=??????��
��−??????��
��2. ??????��
7
49
�
=??????��
749−??????��
7�
3. ??????��
��
�
=�??????��
��3. ??????��
77
5
=5⋅??????��
77
Integrated Basic Education Department | 15
Laws of Logarithms
Laws of Logarithms
Let �>0, �≠1and �∈ℝ.
For �>0, �>0, then: Examples:
1. ??????��
���=??????��
��+??????��
��1. ??????��
327⋅81=??????��
327+??????��
381
2. ??????��
�
�
�
=??????��
��−??????��
��2. ??????��
7
49
�
=??????��
749−??????��
7�
3. ??????��
��
�
=�??????��
��3. ??????��
77
5
=5⋅??????��
77
Notre Dame of Kidapawan College
Integrated Basic Education Department | 16
log(��
2
)
Solution:
log(��
2
)
=loga+logb
2
=loga+2logb
Use the properties/laws of logarithms to expand each
expression in terms of the logarithms of the factors. Assume
each factor is positive.
log
3
3
�
3
Solution:
=3log
3
3
�
=3(log
33−log
3�)
=3(1−log
3�)
=3−3log
3�
Law 1
Law 3
Property 2
Law 2
Law 3
Distributive
Property
Integrated Basic Education Department | 16
log(��
2
)
Solution:
log(��
2
)
=loga+logb
2
=loga+2logb
Use the properties/laws of logarithms to expand each
expression in terms of the logarithms of the factors. Assume
each factor is positive.
log
3
3
�
3
Solution:
=3log
3
3
�
=3(log
33−log
3�)
=3(1−log
3�)
=3−3log
3�
Law 1
Law 3
Property 2
Law 2
Law 3
Distributive
Property
Notre Dame of Kidapawan College
Integrated Basic Education Department | 17
log2+log3
=log2⋅3
=log6
Use the properties/laws of logarithms to
condense the expressions as a single logarithm.
log
5(�
2
)−3log
5�
=log
5
�
2
�
3
=log
5(�
2−3
)
=log
5(�
−1
)
=−log
5�
Integrated Basic Education Department | 17
log2+log3
=log2⋅3
=log6
Use the properties/laws of logarithms to
condense the expressions as a single logarithm.
log
5(�
2
)−3log
5�
=log
5
�
2
�
3
=log
5(�
2−3
)
=log
5(�
−1
)
=−log
5�
Notre Dame of Kidapawan College
Integrated Basic Education Department | 18
Leta,bandxbepositiverealnumberssuchthat�≠
1,b≠1.
log
��=
log��
log��
Change-of-Base Formula
log
832
=
log
232
log28
=
5
3
log
243
1
27
=
log
3
1
27
log
3243
=
−3
5
Integrated Basic Education Department | 18
Leta,bandxbepositiverealnumberssuchthat�≠
1,b≠1.
log
��=
log��
log��
Change-of-Base Formula
log
832
=
log
232
log28
=
5
3
log
243
1
27
=
log
3
1
27
log
3243
=
−3
5
Notre Dame of Kidapawan College
Integrated Basic Education Department | 19
Somestrategiesinsolvinglogarithmicequations:
a.rewritingtoexponentialform
b.usinglogarithmicproperties/laws
c.applyingtheone-to-onepropertyofalogarithmic
functions,iflog
��=log
��,then�=�.
d.applyingtheZeroFactorProperty,ifab=0,then
a=0orb=0.
e.Aftersolving,checkifeachoftheobtained
valuesdoesnotresultinundefinedexpressionsinthe
givenequation.Ifso,thenthesevalueswouldNOTbe
consideredsolutions.
Solving Logarithmic Equations
Integrated Basic Education Department | 19
Somestrategiesinsolvinglogarithmicequations:
a.rewritingtoexponentialform
b.usinglogarithmicproperties/laws
c.applyingtheone-to-onepropertyofalogarithmic
functions,iflog
��=log
��,then�=�.
d.applyingtheZeroFactorProperty,ifab=0,then
a=0orb=0.
e.Aftersolving,checkifeachoftheobtained
valuesdoesnotresultinundefinedexpressionsinthe
givenequation.Ifso,thenthesevalueswouldNOTbe
consideredsolutions.
Solving Logarithmic Equations
Notre Dame of Kidapawan College
Integrated Basic Education Department | 20
Findthevalueofxinthefollowinglogarithmicequations:
log
42�=log
412
2�=12
�=6
log
3(2�−1)=2
2�−1=3
2
2�−1=9
2�=10
�=5
Integrated Basic Education Department | 20
Findthevalueofxinthefollowinglogarithmicequations:
log
42�=log
412
2�=12
�=6
log
3(2�−1)=2
2�−1=3
2
2�−1=9
2�=10
�=5
Notre Dame of Kidapawan College
Integrated Basic Education Department | 21
Giventhelogarithmicexpressionlog
��,
If0<�<1,then�
1<�
2ifandonlyiflog
��
1>
log
��
2.
If�>1,then�
1<�
2ifandonlyiflog
��
1<
log
��
2.
Solving Logarithmic Inequalities
Integrated Basic Education Department | 21
Giventhelogarithmicexpressionlog
��,
If0<�<1,then�
1<�
2ifandonlyiflog
��
1>
log
��
2.
If�>1,then�
1<�
2ifandonlyiflog
��
1<
log
��
2.
Solving Logarithmic Inequalities
Notre Dame of Kidapawan College
Integrated Basic Education Department | 22
Solve the following logarithmic
inequalities:
log
3(2�−1)>log
3(�+2)
2�−1>�+2
2�−�>1+2
�>3or (3,+∞)
log
3�≤5
�≤3
5
�≤243The solution is (0,ሿ243.
Integrated Basic Education Department | 22
Solve the following logarithmic
inequalities:
log
3(2�−1)>log
3(�+2)
2�−1>�+2
2�−�>1+2
�>3or (3,+∞)
log
3�≤5
�≤3
5
�≤243The solution is (0,ሿ243.
Notre Dame of Kidapawan College
Integrated Basic Education Department | 23
Logarithmcanbewritten
inexponentialform.
Infact,theinverse
ofanexponentialfunction
isthelogarithmicfunction.
Table of Values, Graphs, and
Equations of Logarithmic Functions
��=�
�
�=�
�
�=�
�
�=log
��
�
−1
�=log
��
Integrated Basic Education Department | 23
Logarithmcanbewritten
inexponentialform.
Infact,theinverse
ofanexponentialfunction
isthelogarithmicfunction.
Table of Values, Graphs, and
Equations of Logarithmic Functions
��=�
�
�=�
�
�=�
�
�=log
��
�
−1
�=log
��
Notre Dame of Kidapawan College
Integrated Basic Education Department | 24
Representthelogarithmicfunction��=log
2�(base
greaterthan1)throughitstableofvalues,graphand
equation:
�
1
16
1
8
1
4
1
2
1 2 4 8
��-4 -3 -2 -1 0 1 2 3
�(�)=log
2�
Integrated Basic Education Department | 24
Representthelogarithmicfunction��=log
2�(base
greaterthan1)throughitstableofvalues,graphand
equation:
�
1
16
1
8
1
4
1
2
1 2 4 8
��-4 -3 -2 -1 0 1 2 3
�(�)=log
2�
Notre Dame of Kidapawan College
Integrated Basic Education Department | 25
Comparison of the graphs of
????????????=�
??????
and ????????????=log
2�.
�=2
�
�=log
2�
�=�
Integrated Basic Education Department | 25
Comparison of the graphs of
????????????=�
??????
and ????????????=log
2�.
�=2
�
�=log
2�
�=�
Notre Dame of Kidapawan College
Integrated Basic Education Department | 26
Constructatableofvaluesoforderedpairsforthe
logarithmicfunction�=log1
2
�anddrawitsgraph.
�
1
16
1
8
1
4
1
2
1 2 4 8
��4 3 2 1 0 -1 -2 -3
�=log1
2
�
Integrated Basic Education Department | 26
Constructatableofvaluesoforderedpairsforthe
logarithmicfunction�=log1
2
�anddrawitsgraph.
�
1
16
1
8
1
4
1
2
1 2 4 8
��4 3 2 1 0 -1 -2 -3
�=log1
2
�
Notre Dame of Kidapawan College
Integrated Basic Education Department | 27
Thegraphsof�=log
2�and�=log1
2
�indicatethatthegraphsofthe
logarithmicfunction�=log
��dependsonthevalueof�.We
generalizetheresultsasfollows:
Analyzing the logarithmic function
�=log
2�
�=log1
2
�
�=log
��(�>1) �=log
��(0<�<1)
Integrated Basic Education Department | 27
Thegraphsof�=log
2�and�=log1
2
�indicatethatthegraphsofthe
logarithmicfunction�=log
��dependsonthevalueof�.We
generalizetheresultsasfollows:
Analyzing the logarithmic function
�=log
2�
�=log1
2
�
�=log
��(�>1) �=log
��(0<�<1)
Notre Dame of Kidapawan College
Integrated Basic Education Department | 28
i.Thedomainisthesetofallpositivenumbers,
�∈ℝ∣�>0.Recallthatthesearepreciselythe
permittedvaluesof�intheexpressionlog
��.
ii.Therangeisthesetofallpositiverealnumbers.
iii.Itisone-to-onefunction.ItsatisfiestheHorizontal
LineTest.
iv.Thex-interceptis1.thereisnoy-intercept.
v.Theverticalasymptoteistheline�=0(orthey-
axis).Thereisnohorizontalasymptote.
Properties of Logarithmic Functions of
the form �=log
��, (??????>�or �<??????<�)
Integrated Basic Education Department | 28
i.Thedomainisthesetofallpositivenumbers,
�∈ℝ∣�>0.Recallthatthesearepreciselythe
permittedvaluesof�intheexpressionlog
��.
ii.Therangeisthesetofallpositiverealnumbers.
iii.Itisone-to-onefunction.ItsatisfiestheHorizontal
LineTest.
iv.Thex-interceptis1.thereisnoy-intercept.
v.Theverticalasymptoteistheline�=0(orthey-
axis).Thereisnohorizontalasymptote.
Properties of Logarithmic Functions of
the form �=log
��, (??????>�or �<??????<�)
Notre Dame of Kidapawan College
Integrated Basic Education Department | 29
Relationship between Graphs of
Logarithmic and Exponential Function
�=�
�
and �=log
��(�>1) �=�
�
and �=log
��(0<�<1)
�=�
�
�=log
��
�=�
�
�=log
��
Integrated Basic Education Department | 29
Relationship between Graphs of
Logarithmic and Exponential Function
�=�
�
and �=log
��(�>1) �=�
�
and �=log
��(0<�<1)
�=�
�
�=log
��
�=�
�
�=log
��
Notre Dame of Kidapawan College
Integrated Basic Education Department | 30
Graphanddeterminethedomain,range,x-intercept,zero
andverticalasymptoteof�=2log
2�.
The Domain, Range, Intercepts, Zeroes, and
Asymptotes of Logarithmic Function
�
1
16
1
8
1
4
1
2
1248
log
2�-4-3-2-10123
2log
2�-8-6-4-20246
Domain: �∣�∈ℝ,�>0or (0,+∞)
Range: �∣�∈ℝor (−∞,+∞)
Vertical Asymptote: �=0
x-intercept: 1
Zero of a function: 1
log
2�
2log
2�
Integrated Basic Education Department | 30
Graphanddeterminethedomain,range,x-intercept,zero
andverticalasymptoteof�=2log
2�.
The Domain, Range, Intercepts, Zeroes, and
Asymptotes of Logarithmic Function
�
1
16
1
8
1
4
1
2
1248
log
2�-4-3-2-10123
2log
2�-8-6-4-20246
Domain: �∣�∈ℝ,�>0or (0,+∞)
Range: �∣�∈ℝor (−∞,+∞)
Vertical Asymptote: �=0
x-intercept: 1
Zero of a function: 1
log
2�
2log
2�
Notre Dame of Kidapawan College
Integrated Basic Education Department | 31
Graphanddeterminethedomain,range,x-intercept,zero
andverticalasymptoteof�=log
3�−1
Domain: �∣�∈ℝ,�>0or (0,+∞)
Range: �∣�∈ℝor (−∞,+∞)
Vertical Asymptote: �=0
x-intercept: 3
Zero of a function: 3
Note that x-intercept can be obtained
graphically and algebraically by setting �=0
�139
log
3�012
log
3�−1-101
log
3�
log
3�−1
Integrated Basic Education Department | 31
Graphanddeterminethedomain,range,x-intercept,zero
andverticalasymptoteof�=log
3�−1
Domain: �∣�∈ℝ,�>0or (0,+∞)
Range: �∣�∈ℝor (−∞,+∞)
Vertical Asymptote: �=0
x-intercept: 3
Zero of a function: 3
Note that x-intercept can be obtained
graphically and algebraically by setting �=0
�139
log
3�012
log
3�−1-101
log
3�
log
3�−1
Notre Dame of Kidapawan College
Integrated Basic Education Department | 32
Graphof��=�⋅??????��
��−�+�
i.Thevalueof�(either�>1or0<�<1)
determineswhetherthegraphisincreasingor
decreasing.
ii.Thevalueof�determinesthestretchorshrinkingof
thegraph.Further,if�isnegative,thereisa
reflectionofthegraphaboutx-axis.
iii.Basedon��=�⋅??????��
��theverticalshiftis�units
up(if�>0)or�unitsdown(if�<0),andthe
horizontalshiftis�unitstotheright(ifc>0)or�units
totheleft(ifc>0).
General Guidelines for Graphing
Transformations of Logarithmic Functions
Integrated Basic Education Department | 32
Graphof��=�⋅??????��
��−�+�
i.Thevalueof�(either�>1or0<�<1)
determineswhetherthegraphisincreasingor
decreasing.
ii.Thevalueof�determinesthestretchorshrinkingof
thegraph.Further,if�isnegative,thereisa
reflectionofthegraphaboutx-axis.
iii.Basedon��=�⋅??????��
��theverticalshiftis�units
up(if�>0)or�unitsdown(if�<0),andthe
horizontalshiftis�unitstotheright(ifc>0)or�units
totheleft(ifc>0).
General Guidelines for Graphing
Transformations of Logarithmic Functions
Notre Dame of Kidapawan College
Integrated Basic Education Department | 33
��=2??????��
Τ
1
2
(�+2)
�-2−
3
2
-1 0 1 2 4 6
��E 2 0 -2-3.17-4-5.17-6
Domain: (−2,+∞)
Range: (−∞,+∞)
Vertical Asymptote: �=−2
x-intercept: -1
y-intercept: -2
Zero of a function: -1
Integrated Basic Education Department | 33
��=2??????��
Τ
1
2
(�+2)
�-2−
3
2
-1 0 1 2 4 6
��E 2 0 -2-3.17-4-5.17-6
Domain: (−2,+∞)
Range: (−∞,+∞)
Vertical Asymptote: �=−2
x-intercept: -1
y-intercept: -2
Zero of a function: -1
Notre Dame of Kidapawan College
Integrated Basic Education Department | 34
EarhquakeMagnitudeonaRichterScale
Themagnitude??????ofanearthquakeisgivenby,
??????=
2
3
log
??????
10
4.40
where??????(injoules)istheenergyreleasedbythe
earthquake(thequantity10
4.40
istheenergyreleasedbya
verysmallreferenceearthquake).
Representation of Real-life
Situation that Use Logarithms
Integrated Basic Education Department | 34
EarhquakeMagnitudeonaRichterScale
Themagnitude??????ofanearthquakeisgivenby,
??????=
2
3
log
??????
10
4.40
where??????(injoules)istheenergyreleasedbythe
earthquake(thequantity10
4.40
istheenergyreleasedbya
verysmallreferenceearthquake).
Representation of Real-life
Situation that Use Logarithms
Notre Dame of Kidapawan College
Integrated Basic Education Department | 35
Supposethatanearthquakereleasedapproximately10
12
joulesofenergy.
(a)WhatisitsmagnitudeonaRichterscale?
(b)Howmuchmoreenergydoesthisearthquakereleasethanthatbythe
referenceearthquake?
Example
(a)Since ??????=10
12
, then ??????=
2
3
log
??????
10
4.40
??????=
2
3
log
10
12
10
4.40
??????=
2
3
log10
7.6
Since by the definition, log10
7.6
=7.6
??????=
2
3
(7.6)
??????≈5.1
(b) The earthquake releases
10
12
10
4.40
=10
7.6
10
7.6
≈39,810,717joules
Integrated Basic Education Department | 35
Supposethatanearthquakereleasedapproximately10
12
joulesofenergy.
(a)WhatisitsmagnitudeonaRichterscale?
(b)Howmuchmoreenergydoesthisearthquakereleasethanthatbythe
referenceearthquake?
Example
(a)Since ??????=10
12
, then ??????=
2
3
log
??????
10
4.40
??????=
2
3
log
10
12
10
4.40
??????=
2
3
log10
7.6
Since by the definition, log10
7.6
=7.6
??????=
2
3
(7.6)
??????≈5.1
(b) The earthquake releases
10
12
10
4.40
=10
7.6
10
7.6
≈39,810,717joules
Notre Dame of Kidapawan College
Integrated Basic Education Department | 36
Exponential Form Logarithmic Form
1. 2
4
=16
2. 11
2
=�
3. 7
�
=400
4. 4=log
8�
5. 3=log
�27
ASSESSMENT
Solve the logarithmic equation:
•log
��=2log
�3+log
�5
Solve the logarithmic inequality and identify the solution set:
•log
2(�−7)<3
Integrated Basic Education Department | 36
Exponential Form Logarithmic Form
1. 2
4
=16
2. 11
2
=�
3. 7
�
=400
4. 4=log
8�
5. 3=log
�27
ASSESSMENT
Solve the logarithmic equation:
•log
��=2log
�3+log
�5
Solve the logarithmic inequality and identify the solution set:
•log
2(�−7)<3
Notre Dame of Kidapawan College
Integrated Basic Education Department | 37
ASSESSMENT
��=??????��
2(�+4)
Do the following:
1. Set up a table of values
2. Draw the graph of the function
3. Determine the sets of domain and range
4. Find the asymptotes
5. Find the x-and y-intercepts
Integrated Basic Education Department | 37
ASSESSMENT
��=??????��
2(�+4)
Do the following:
1. Set up a table of values
2. Draw the graph of the function
3. Determine the sets of domain and range
4. Find the asymptotes
5. Find the x-and y-intercepts
Notre Dame of Kidapawan College
Integrated Basic Education Department | 38
Do you have
clarifications about
the discussion?
Integrated Basic Education Department | 38
Do you have
clarifications about
the discussion?
Notre Dame of Kidapawan College
Integrated Basic Education Department | 39
References
‣Dimasuay,L.,Alcala,J.,&Palacio,J.(2021).
GeneralMathematicsforSeniorHighSchool.
Quezon:C&EPublishing.
‣Versoza,D.M.,A,C.L.,Hao,L.C.,Miro,E.
D.,Ocampo,S.R.,Palomo,E.G.,&
Tresvalles,R.M.(2016).TeachingGuidefor
SeniorHighSchoolGeneralMathematics.
Quezon:CommissiononHigherEducation.
Integrated Basic Education Department | 39
References
‣Dimasuay,L.,Alcala,J.,&Palacio,J.(2021).
GeneralMathematicsforSeniorHighSchool.
Quezon:C&EPublishing.
‣Versoza,D.M.,A,C.L.,Hao,L.C.,Miro,E.
D.,Ocampo,S.R.,Palomo,E.G.,&
Tresvalles,R.M.(2016).TeachingGuidefor
SeniorHighSchoolGeneralMathematics.
Quezon:CommissiononHigherEducation.
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