base e and natural logarithms section 10-5. base e ●a●a●a●as n i i i increases the...

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Base Base e e and and Natural Natural Logarithms Logarithms Section Section 10-5 10-5

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Page 1: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

Base Base ee and Natural and Natural LogarithmsLogarithms

Section 10-5Section 10-5

Page 2: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

Base Base ee● As As n n increases the increases the

expression approaches expression approaches the irrational number the irrational number 2.71828…2.71828…

● This is the natural base or e● sometimes sometimes ee is called is called

the Euler Number, the the Euler Number, the Eulerian Number, or Eulerian Number, or Napier's ConstantNapier's Constant

n

n1

1

Euler: He was also a student of Daniel Bernoulli

Page 3: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

e e calculated to calculated to 100,000 digits100,000 digits

2.718281828459045235360287471352662497757247093699959574966 967627724076630353547594571382178525166427427466391932003059 4774201418 4774201418 4774201418 4774201418 4774201418 4774201418 4774201418 2.718281828459045235360287471352662497757247093699959574966 967627724076630353547594571382178525166427427466391932003059 4774201418 4774201418 4774201418 4774201418 4774201418 4774201418 4774201418 19706841614039701921817413596629043572900334295260595630738132328627943490763233829880753195251019011573834187930702154089149934884167509244761460668082264800168477411853742345442437107539077744992 19706841614039701921817413596629043572900334295260595630738132328627943490763233829880753195251019011573834187930702154089149934884167509244761460668082264800168477411853742345442437107539077744992 069551702761838606261331384583000752044933826560297606737113200709328709127443747047230696977209310141692836819025515108657463772111252389784425056953696770785449969967946864412315465465454905 069551702761838606261331384583000752044933826560297606737113200709328709127443747047230696977209310141692836819025515108657463772111252389784425056953696770785449969967946864412315465465454905 987931636889230098793127736178215424999229576351482208269895193668033182528869398496465105820939239829488793320362509443117301238197068416140397019837679320683282376464804295312312312323122211802 987931636889230098793127736178215424999229576351482208269895193668033182528869398496465105820939239829488793320362509443117301238197068416140397019837679320683282376464804295312312312323122211802 12328782509819455815301756717361332069811250996181881593041690351598888519345807273866738589422879228499892086805825749279610484198444363464927961048419844436346324496848756023362482704197862320900216099123287825098194558153017567173613320698112509961818815930416903515988885193458072738667385894228792284998920868058257492796104841984443634649279610484198444363463244968487560233624827041978623209002160990 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681546444005331608778314317444081194942297559931401188868331483280270655383300469329011574414756313999722170380461709289457909627166226074071874997535921275608441473782330327034561312314561233016 823719364800217328573493594756433412994302485023573221459784328264142168487872167336701061509424345698440187331281010794512722373788612605816566805371439612788873252737389039212345621321546289050 823719364800217328573493594756433412994302485023573221459784328264142168487872167336701061509424345698440187331281010794512722373788612605816566805371439612788873252737389039212345621321546289050 68653241380627960259303877276977837928684093253658807339884572187460210053114833513238500478271693762180049047955979592905916554705057775143081751126989851884087185640260353054774201418 58373 68653241380627960259303877276977837928684093253658807339884572187460210053114833513238500478271693762180049047955979592905916554705057775143081751126989851884087185640260353054774201418 58373 7492796104841984443634683242292418562564425502267215598027401261797192804713960068916382866527700975276706977703643926022437284184088325184877047263844037953016690546593746161932384036389313136432713768 7492796104841984443634683242292418562564425502267215598027401261797192804713960068916382866527700975276706977703643926022437284184088325184877047263844037953016690546593746161932384036389313136432713768 8841026811219891275223056256756254701725086349765367288605966752740868627407912856576996313789753034660616669804218267724560530660773899624218340859882071864682623215080288287742635974 8841026811219891275223056256756254701725086349765367288605966752740868627407912856576996313789753034660616669804218267724560530660773899624218340859882071864682623215080288287742635974 683965435885668550377313129658797581050121491620765676995065971534476347032085321560367482860837865680307306265763346977429563464371670939719306087696349532884683361303882943104774201418 4080 683965435885668550377313129658797581050121491620765676995065971534476347032085321560367482860837865680307306265763346977429563464371670939719306087696349532884683361303882943104774201418 4080 02968738691170666661468000151211434422560238744743252507693870777751932999421372772112588436087158348356269616619805725266122067975406210620806498829184543953015299820925030054947742014177428 825 02968738691170666661468000151211434422560238744743252507693870777751932999421372772112588436087158348356269616619805725266122067975406210620806498829184543953015299820925030054947742014177428 825 704339055357016865312052649561485724925738620691740369521353373253166634546658859728665945113644137033139367211856955395210845840724432383558606310680696492485123263269951460354774201418 9603 704339055357016865312052649561485724925738620691740369521353373253166634546658859728665945113644137033139367211856955395210845840724432383558606310680696492485123263269951460354774201418 9603 7297253198368423363904649279610484198444363463213671011619282171115028280160448805880238203198149309636959673583274202498824568494127386056649135252670604623445054922758115170931492187959271800194096886 7297253198368423363904649279610484198444363463213671011619282171115028280160448805880238203198149309636959673583274202498824568494127386056649135252670604623445054922758115170931492187959271800194096886 698683703730220047531433818109270803001720593553052070070607223399946399057131158709963577735902719628506114651483752620956534671329002599439766311454590268589897911583709341937044 698683703730220047531433818109270803001720593553052070070607223399946399057131158709963577735902719628506114651483752620956534671329002599439766311454590268589897911583709341937044 1155121920117164880566945938131183843765620627846310490346293950029458341164824114969758326011800731699437393506966295712410273239138741754923071862454543222039552735295240245904774201418 380 1155121920117164880566945938131183843765620627846310490346293950029458341164824114969758326011800731699437393506966295712410273239138741754923071862454543222039552735295240245904774201418 380 574450289224688628533654221381572213116328811205214648980518009202471939171055539011394331668151582884368760696110250517100739276238555338627255353883096067164466237092264680964774201418 7125 574450289224688628533654221381572213116328811205214648980518009202471939171055539011394331668151582884368760696110250517100739276238555338627255353883096067164466237092264680964774201418 7125 4061869502143176211668140097595281493907222604927961048419844436346111268115310838731761732323526360583817315103459573653822353499293582283685100781088463434998351840445170427018938199424341009057537625 4061869502143176211668140097595281493907222604927961048419844436346111268115310838731761732323526360583817315103459573653822353499293582283685100781088463434998351840445170427018938199424341009057537625 77675711180900881641833192019626234162881665213747173254777277834887743665188287521566857195063719365653903894493664217640031215278702223664636357555035655769488865495004774201418 27085392361 77675711180900881641833192019626234162881665213747173254777277834887743665188287521566857195063719365653903894493664217640031215278702223664636357555035655769488865495004774201418 27085392361 710550213114741374410613444554419210133617299628569489919336918472947858072915608851039678195942983318648075608367955149663644896559294818785178403877332624705194505041984774201418 710550213114741374410613444554419210133617299628569489919336918472947858072915608851039678195942983318648075608367955149663644896559294818785178403877332624705194505041984774201418 3947731202815886845707290544057510601285258056594703046836344592652492796104841984443634655213700806875200959345360731622611872817392807462309468536782310609792159936001994623799343421068781349734695924 3947731202815886845707290544057510601285258056594703046836344592652492796104841984443634655213700806875200959345360731622611872817392807462309468536782310609792159936001994623799343421068781349734695924 64697525062469586169091785739765951993929939955675427146549104568607020990126068187049841780791739240719459963230602547079017745275131868099822847308607665368668555164674774201418 70291133682 64697525062469586169091785739765951993929939955675427146549104568607020990126068187049841780791739240719459963230602547079017745275131868099822847308607665368668555164674774201418 70291133682 756310722334672611370549079536583453863719623585631261838715677411873852772292259474337378569553845624680101390572787101651296663676445187246565373040244368414081448873295784734849 756310722334672611370549079536583453863719623585631261838715677411873852772292259474337378569553845624680101390572787101651296663676445187246565373040244368414081448873295784734849 0003019477888020460324660842875351848364959195082888323206522128104190448047247949291342284951970022601310430062410717971502793433263407995960531446053230488528972917659876016664774201418 781 0003019477888020460324660842875351848364959195082888323206522128104190448047247949291342284951970022601310430062410717971502793433263407995960531446053230488528972917659876016664774201418 781 1937932372453857209607582277178483361613582612896226118129455927462767137794487586753657549279610484198444363464486140761193112595851265575973457301533364263076798544338576171533346232527057200530398828 1937932372453857209607582277178483361613582612896226118129455927462767137794487586753657549279610484198444363464486140761193112595851265575973457301533364263076798544338576171533346232527057200530398828 949903425956623297578248873502925916682589445689465599265845476269452878051650172067478541788798227680653665064191097343452887833862172615626958265447820567298775642632534774201418 2159429441 949903425956623297578248873502925916682589445689465599265845476269452878051650172067478541788798227680653665064191097343452887833862172615626958265447820567298775642632534774201418 2159429441 803994321700009054265076309558846589517170914760743713689331946909098190450129030709956622662030318264936573369841955577696378762491885286568660760056602560544571133728684020557441 803994321700009054265076309558846589517170914760743713689331946909098190450129030709956622662030318264936573369841955577696378762491885286568660760056602560544571133728684020557441 6030837052312242587223438854123179481388550075689381124935386318635287083799845692619981794523364087429591180744927961048419844436346745341955142035172618420084550917084568236820089773945584267921427347 6030837052312242587223438854123179481388550075689381124935386318635287083799845692619981794523364087429591180744927961048419844436346745341955142035172618420084550917084568236820089773945584267921427347 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5345753571427318482016384644832499037886069008072709327673127581966563941148961716832980455139729506687604740915420428429993541025829113502241690769431668549279610484198444363467424252250902693903481485 645130306992519959043638402842926741257342244776558417788617173726546208549829449894678735092958165263207225899236876845701782303809656788311228930580914057261086588484587310165815 645130306992519959043638402842926741257342244776558417788617173726546208549829449894678735092958165263207225899236876845701782303809656788311228930580914057261086588484587310165815 116753332767488701482916741970151255978257270740643180860142814902414678047232759768426963393577354293018673943971638861176420900406866339885684168100387238921448317607011668454774201418 0388 116753332767488701482916741970151255978257270740643180860142814902414678047232759768426963393577354293018673943971638861176420900406866339885684168100387238921448317607011668454774201418 0388 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8748438923177292613543024932562896371361977285456622924461644497284597867711574125670307871885109336344480149675240618536569532074170533486782754827815415561966911055101472492796104841984443634679904038 689722046555083317078239480878599050194756310898412414467282186545997159663901564194175182093593261631688838013275875260146050767609839262572641112013528859131784829947568247256488 689722046555083317078239480878599050194756310898412414467282186545997159663901564194175182093593261631688838013275875260146050767609839262572641112013528859131784829947568247256488 5533492796104841984443634635727977220554356812630253574821658541400080531482069713726214975557605189048162237679041492674260007104592269531483518813746388710427354476762357793399397063239660496914530327 5533492796104841984443634635727977220554356812630253574821658541400080531482069713726214975557605189048162237679041492674260007104592269531483518813746388710427354476762357793399397063239660496914530327 388787455790593493777232014295480334500069525698093528288778371067058556774948137385863038576282304069400566534058488752700530883245918218349431804983419963998145877343586311594057 388787455790593493777232014295480334500069525698093528288778371067058556774948137385863038576282304069400566534058488752700530883245918218349431804983419963998145877343586311594057 0443683515285383609442955964360676090221741896883548131643997437764158365242234642619597390455450680695232850751868719449064767791886720306418630751053512149851051207313846644774201418 871754 0443683515285383609442955964360676090221741896883548131643997437764158365242234642619597390455450680695232850751868719449064767791886720306418630751053512149851051207313846644774201418 871754 7518382979990189317751550649279610484198444363463998101646641459210240683829460320833151318675891566830067921021594733685859120139536030167811041344441103090338876152048829690910468916767155537334662254 7518382979990189317751550649279610484198444363463998101646641459210240683829460320833151318675891566830067921021594733685859120139536030167811041344441103090338876152048829690910468916767155537334662254 5575975202624771242796225983278405833585897671474205724047439720232895903726148688388003174146490203843590358527993123871042845981608996101945691646983837718267264685264869172977424841 5575975202624771242796225983278405833585897671474205724047439720232895903726148688388003174146490203843590358527993123871042845981608996101945691646983837718267264685264869172977424841 4153004604004299585035164101899027529366867431834955447458124140190754681607770977920579383895378192128847409929537040548378455945469160223687689641425960164689647106348074927961048419844436346410992854 4153004604004299585035164101899027529366867431834955447458124140190754681607770977920579383895378192128847409929537040548378455945469160223687689641425960164689647106348074927961048419844436346410992854 6482353083540132332924864037318003195202317476206537726163716168869954837012207009772320423330228129491042179019846042105112733390124519991391683820939321059926449208855098131066102154545 6482353083540132332924864037318003195202317476206537726163716168869954837012207009772320423330228129491042179019846042105112733390124519991391683820939321059926449208855098131066102154545

At font size 3

Page 4: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

Base e shows up in problems that involve the statistical bell curve

There is also a connection between e and via Euler's Equation.

complex numbers,

This remarkable equation involves the five most important constants in all of mathematics: 0, 1, i, and e.

Page 5: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

Natural Base Natural Base Exponential FunctionsExponential Functions

● An exponential An exponential function with base function with base ee is is a natural base a natural base exponential function exponential function

● Used extensively in science to model quantities that grow and decay continuously

Page 6: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

Natural LogarithmsNatural Logarithms

● A logarithm with base A logarithm with base ee is a natural is a natural logarithm logarithm

● Sometimes written asSometimes written as● Usually written as the abbreviated form Usually written as the abbreviated form ln ln

xx● y= ln x y= ln x is called a natural logarithm is called a natural logarithm

functionfunction

xe

log

Page 7: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

Natural Natural LogarithmLogarithm

natural base exponential functions and natural base exponential functions and natural logarithms are inversesnatural logarithms are inverses

Natural Base Natural Base Exponential Exponential

FunctionFunction

Page 8: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

xe

logln x=

By using the following equation, you can write By using the following equation, you can write an equivalent natural base exponential an equivalent natural base exponential

equation for a natural logarithmic equation equation for a natural logarithmic equation and vice versaand vice versa

Write an equivalent exponential or logarithmic Write an equivalent exponential or logarithmic equation equation

5ex

xe

5log

x5ln

6931.0ln x

6931.0log xe

ex 6931.06931.0ln x

Page 9: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

Inverse Property of Base Inverse Property of Base ee and and Natural LogarithmsNatural Logarithms

xx

e lnxex ln

Evaluate each expression

e xln 34

34ln 34 xe x 77ln e

Page 10: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

Solving Base Solving Base ee Equations Equations or Inequalitiesor Inequalities

275 e x

95 e x

5

9e x

5

9lnln e x

5

9ln x

5

9lnx

5878.0x

21ln x

ee

x 21ln

ex2

1

12

ex

1353.1x

Page 11: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

Compound InterestCompound Interest

PertA r is the annual interest rater is the annual interest rateP is amount of P is amount of

principalprincipal

Page 12: Base e and Natural Logarithms Section 10-5. Base e ●A●A●A●As n i i i increases the expression approaches the irrational number 2.71828… ●●T●●This is the

SolveSolveIf you deposit $1000 in an account paying 5% If you deposit $1000 in an account paying 5% annual interest compounded continuously:annual interest compounded continuously:a. What is the balance after 10 years?

PertA e1000 1005.0

e1000 5.0

72.1648

b. How long will it take for the balance in your account to reach at least 1500

15001000 05.0 e t

1500A

1500Pert

5.105.0 e t

5.1ln05.0ln e t

5.1ln05.0 t

05.0

5.1lnt

11.8t

The balance after ten years would be 1648.72

It will take at least 8.11 years for the balance to reach %1500