1. Base e and Natural Logarithms Section 10-5

2. 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 or e ●s●s●s●sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant Euler: He was also a student of Daniel Bernoulli

3. e calculated to 100,000 digits 2.718281828459045235360287471352662497757247093699959574966 967627724076630353547594571382178525166427427466391932003059 4774201418 4774201418 4774201418 4774201418 4774201418 4774201418 4774201418 197068416140397019218174135966290435729003342952605956307381323286279434907632338298807531952510190115738341879307021540891499348841675092447614606680822648001684774118537423454424371075390777449 92 069551702761838606261331384583000752044933826560297606737113200709328709127443747047230696977209310141692836819025515108657463772111252389784425056953696770785449969967946864412315465465454905 987931636889230098793127736178215424999229576351482208269895193668033182528869398496465105820939239829488793320362509443117301238197068416140397019837679320683282376464804295312312312323122211802 123287825098194558153017567173613320698112509961818815930416903515988885193458072738667385894228792284998920868058257492796104841984443634649279610484198444363463244968487560233624827041978623209 002160990 23530436994184914631409343173814364054625315209618369088870701676839642437814059271456354906130310720851038375051011574770417189861068739696552126715468895703503540212340784981245419334 321068170121005627880235193033224745015853904730419957777093503660416997329725088687696640355570716226844716256079882651787134195124665201030592123667719432527867539855894489697231546791234655096 409754591856956380236370162112047742722836489613422516445078182442352948636372141740238893441247963574370263755294448337998016125492278509257782562092622648326277933386566123123123123123481627725 164019105900491644998289315056604725802778631864155195653244258698294695930801915298721172556347546396447910145904090586298496791249279610484198444363468740687050489585867174798546677575732056812 8845920 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330535370876138939639177957454016137223618789365260538155841587186925538606164779834025435128439612946035291332594279490433729908573158029095863138268324927961048419844436346914771163963370924003 1689458 6360606458459251269946557248391865642097526850823075442545993769170419777800853627309417101634349076964237222943523661255725088147792231519747780605696725380171807763603462459271123123877 84658506560507808442115296975218908740196609066518035165017925046195013665854366327125496399085491442000145747608193022120660243300964127048943903971771951806990869986066365832311111111111111227 870937650226014929101151717763594460202324930028040186772391028809786660565118326004368850881715723866984224220102495055188169480322100251542649463981287367765892768816456456455454555359831247788 652014117411091360116499507662907794364600585194199856016264790761532103872755712699251827568798930276176114616254935649590379804583818232336861201624373656984670378585330527583332465644513878953 793990752166069238053369887956513728559388349989470741618155012539706464817194670834819721448889879067650379590366967249499254527903372963616265897603949857674139735944102374492796104841984443634 6432970 9355477982629614591442936451428617158587339746791897571211956187385783644758448423555581050025611492391518893099463428413936080383091662818811503715284967059741625628236092168122122207515 017772538740256425347087908913729172282861151591568372524163077225440633787593105982676094420326192428531701878177296023541306067213604600038966109364709514141718577701418061231231232123210644363 681546444005331608778314317444081194942297559931401188868331483280270655383300469329011574414756313999722170380461709289457909627166226074071874997535921275608441473782330327034561312314561233016 823719364800217328573493594756433412994302485023573221459784328264142168487872167336701061509424345698440187331281010794512722373788612605816566805371439612788873252737389039212345621321546289050 68653241380627960259303877276977837928684093253658807339884572187460210053114833513238500478271693762180049047955979592905916554705057775143081751126989851884087185640260353054774201418 58373 749279610484198444363468324229241856256442550226721559802740126179719280471396006891638286652770097527670697770364392602243728418408832518487704726384403795301669054659374616193238403638931313643 2713768 8841026811219891275223056256756254701725086349765367288605966752740868627407912856576996313789753034660616669804218267724560530660773899624218340859882071864682623215080288287742635974 683965435885668550377313129658797581050121491620765676995065971534476347032085321560367482860837865680307306265763346977429563464371670939719306087696349532884683361303882943104774201418 4080 02968738691170666661468000151211434422560238744743252507693870777751932999421372772112588436087158348356269616619805725266122067975406210620806498829184543953015299820925030054947742014177428 825 704339055357016865312052649561485724925738620691740369521353373253166634546658859728665945113644137033139367211856955395210845840724432383558606310680696492485123263269951460354774201418 9603 729725319836842336390464927961048419844436346321367101161928217111502828016044880588023820319814930963695967358327420249882456849412738605664913525267060462344505492275811517093149218795927180019 4096886 698683703730220047531433818109270803001720593553052070070607223399946399057131158709963577735902719628506114651483752620956534671329002599439766311454590268589897911583709341937044 1155121920117164880566945938131183843765620627846310490346293950029458341164824114969758326011800731699437393506966295712410273239138741754923071862454543222039552735295240245904774201418 380 574450289224688628533654221381572213116328811205214648980518009202471939171055539011394331668151582884368760696110250517100739276238555338627255353883096067164466237092264680964774201418 7125 406186950214317621166814009759528149390722260492796104841984443634611126811531083873176173232352636058381731510345957365382235349929358228368510078108846343499835184044517042701893819942434100905 7537625 77675711180900881641833192019626234162881665213747173254777277834887743665188287521566857195063719365653903894493664217640031215278702223664636357555035655769488865495004774201418 27085392361 710550213114741374410613444554419210133617299628569489919336918472947858072915608851039678195942983318648075608367955149663644896559294818785178403877332624705194505041984774201418 394773120281588684570729054405751060128525805659470304683634459265249279610484198444363465521370080687520095934536073162261187281739280746230946853678231060979215993600199462379934342106878134973 4695924 64697525062469586169091785739765951993929939955675427146549104568607020990126068187049841780791739240719459963230602547079017745275131868099822847308607665368668555164674774201418 70291133682 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4. 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.

5. Natural Base Exponential Functions ●A●A●A●An exponential function with base e is a natural base exponential function ●●U●●Used extensively in science to model quantities that grow and decay continuously

6. Natural Logarithms ● A logarithm with base e is a natural logarithm ● Sometimes written as ● Usually written as the abbreviated form ln x ● y= ln x is called a natural logarithm function

7. Natural Logarithm natural base exponential functions and natural logarithms are inverses Natural Base Exponential Function

8. ln x= By using the following equation, you can write an equivalent natural base exponential equation for a natural logarithmic equation and vice versa Write an equivalent exponential or logarithmic equation

9. Inverse Property of Base e and Natural Logarithms Evaluate each expression

10. Solving Base e Equations or Inequalities

11. Compound Interest r is the annual interest rate P is amount of principal

12. Solve If you deposit $1000 in an account paying 5% annual interest compounded continuously: a. What is the balance after 10 years? b. How long will it take for the balance in your account to reach at least 1500 The balance after ten years would be 1648.72 It will take at least 8.11 years for the balance to reach %1500