5.2. The Natural Logarithm. Solution

Transcription

1 5.2 The Naural Logarihm The number e is an irraional number, similar in naure o π. Is non-erminaing, non-repeaing value is e Like π, e also occurs frequenly in naural phenomena. In fac, of he hree mos commonly used bases in exponenial funcions 2, e, and 10 e is used mos ofen. Why is his? Wha could his unusual number have o do wih such hings as a bacerial culure? The symbols e and π are boh examples of ranscendenal numbers : real numbers ha canno be roos of a polynomial equaion wih ineger coefficiens. As i urns ou, e has some ineresing properies. For example, he insananeous rae of change of y e x as a funcion of x produces he exac same graph; herefore, every higher-level derivaive of y e x also produces he same graph. Example 1 The Naural Logarihm a) Graph he funcion y e x and is inverse using echnology. b) Idenify he key feaures of he graphs. Soluion a) Use graphing echnology. Mehod 1: Use The Geomeer s Skechpad Plo he funcion y e x. Technology Tip You can access he value e from he Values menu in he New Funcion dialogue box. 5.2 The Naural Logarihm MHR 259

2 Consruc a poin on he graph and measure is abscissa (x-coordinae) and ordinae (y-coordinae). Technology Tip You can graph a poin on he inverse funcion by reversing he roles of x and y. Selec he graphs, and plo and race he inverse. Click on he ordinae (y) and abscissa (x), in order. From he Graph menu, choose Plo as (x, y). From he Display menu, choose Trace Ploed Poin. Click and drag he poin on he funcion plo o race ou he inverse of y e x. 260 MHR Calculus and Vecors Chaper 5

3 Mehod 2: Use a Graphing Calculaor Graph he funcion y e x. The inverse of he exponenial funcion is he logarihmic funcion. The log funcion on he graphing calculaor graphs logarihmic funcions wih base 10. To graph y log e x, rewrie i in erms of base 10: y loge x log x log e CONNECTIONS Recall ha log a logc b b =. log a c Graph his funcion. Noice ha hese graphs are reflecions of each oher in he line y x, confirming ha hey are indeed inverse funcions of each oher. The logarihmic funcion having base e occurs very frequenly and has a special name. The naural logarihm of x is defined as ln x log e x. The lef side of his equaion is he naural logarihm of x, and i is read as lon x. You can confirm ha y ln x is he same funcion as y log e x by graphing boh funcions ogeher, using differen line syles: CONNECTIONS Naural logarihms are also someimes called Naperian logarihms, named afer he Scoish mahemaician and philosopher John Napier ( ). b) The following able liss he key feaures of each graph. y e x y ln x Domain: x Domain: {x x,x 0} Range: {y y,y 0} Range: y Increasing on is domain Increasing on is domain y-inercep 1 No y-inercep No x-inercep x-inercep 1 Napier is also famous for invening he decimal poin, as well as a very primiive form of mechanical calculaor. Horizonal asympoe a y 0 (x-axis) No minimum or maximum poin No poin of inflecion Concave up on is domain Verical asympoe a x 0 (y-axis) No minimum or maximum poin No poin of inflecion Concave down on is domain 5.2 The Naural Logarihm MHR 261

4 Example 2 Evaluae e x Evaluae, correc o hree decimal places. a) e 3 b) e 1 2 Soluion Use a scienific or graphing calculaor o find an accurae value. These calculaors have a dedicaed buon for e. a) e b) e 2 e Example 3 Evaluae ln x Evaluae, correc o wo decimal places. a) ln 10 b) ln (5) c) ln e Soluion Scienific and graphing calculaors have a LN key ha can be used o evaluae naural logarihms. a) ln b) ln (5) is undefined. Recall ha he domain of all logarihmic funcions, y 2 including y ln x, is {x x, x 0}, so naural logarihms can only be found for posiive numbers. c) Recall ha ln e log e e and log b b x x Therefore, ln e 1. 2 x ln e x x and e ln x x These properies are useful when you are solving equaions involving exponenial and logarihmic funcions. 262 MHR Calculus and Vecors Chaper 5

5 Example Bacerial Growh The populaion of a bacerial culure as a funcion of ime is given by he equaion P() 200e 0.09, where P is he populaion afer days. a) Wha is he iniial populaion of he bacerial culure? b) Esimae he populaion afer 3 days. c) How long will he bacerial culure ake o double is populaion? d) Rewrie his funcion as an exponenial funcion having base 2. Soluion a) To deermine he iniial populaion, se 0. P() 0 200e0. 09( 0) 200e0 200() The iniial populaion is 200. b) Se 3 o deermine he populaion afer 3 days. P() 3 200e0. 09( 3) 200e ( ) 265 Afer 3 days, he bacerial culure will have a populaion of approximaely 265. c) To find he ime required for he populaion o double, deermine when P() e 0.09 Mehod 1: Use Graphical Analysis Graph he funcion using graphing echnology o idenify he value of when P() 00. The graph shows ha he populaion will double afer abou 7. days. 5.2 The Naural Logarihm MHR 263

6 Mehod 2: Use Algebraic Reasoning Use naural logarihms o solve his equaion algebraically e e0. 09 ln2 lne0. 09 Take he naural logarihm of bohsides. ln ln ex x ln Therefore, he bacerial culure will double afer approximaely 7. days. d) Since he bacerial culure has an iniial populaion of 200 and doubles afer 7. days, he relaionship beween populaion, P, and ime,, can be approximaed by he funcion P () ( 200) 27. Noe ha expresses ime in erms of he number of doubling periods. 7. << >> KEY CONCEPTS The value of e, correc o 12 decimal places, is e ln x log e x The funcions y ln x and y e x are inverses. Many naurally occurring phenomena can be modelled using base-e exponenial funcions. Communicae Your Undersanding C1 How can you deermine he inverse of he funcion y e x a) graphically? b) algebraically? C2 Wha is unique abou he funcion f (x) e x compared o exponenial funcions having bases oher han e? C3 The following wo equaions were used in Example : P() 200e 0.09 P () ( 200) 27. where P represens a populaion of baceria afer days. Why do hese wo funcions yield slighly differen resuls? 26 MHR Calculus and Vecors Chaper 5

7 A Pracise Use his informaion o answer quesions 1 o 3. Le f (x) e x and g(x) ln x. 1. a) Use echnology o graph f (x). b) Idenify he following key feaures of he graph. i) domain ii) range iii) any x- or y-inerceps iv) he equaions of any asympoes v) inervals for which he funcion is increasing or decreasing vi) any minimum or maximum poins vii) any inflecion poins 2. Repea quesion 1 for g(x). 3. Are f (x) and g(x) inverse funcions? Jusify your answer wih mahemaical reasoning.. Esimae he value of each exponenial funcion, wihou using a calculaor. a) e b) e 5 c) e 2 d) e 2 5. Evaluae each expression in quesion, correc o hree decimal places, using a calculaor. 6. Evaluae, if possible, correc o hree decimal places, using a calculaor. a) ln 7 b) ln 200 c) ln 1 d) ln () 7. Wha is he value of ln 0? Why is his reasonable? B Connec and Apply 8. Simplify. a) ln (e 2x ) b) ln (e x ) ln (e x ) c) eln (x1) d) (e ln (3x) )(ln(e 2x )) 9. Solve for x, correc o hree decimal places. a) e x 5 x b) e c) ln (e x ) 0.2 d) eln (2x) 10. a) Solve 3 x 15 by aking he naural logarihm of boh sides. b) Solve 3 x 15 by aking he common logarihm (base 10) of boh sides. c) Wha do you conclude? 11. Chaper Problem Sheona s supervisor has given her some capaciors o analyse. When one of he charged capaciors is conneced o a resisor o form an RC (resisor-capacior) circui, he capacior discharges according o he equaion V() Vmaxe, where V is he volage, in vols; is ime, in seconds; and V max is he iniial volage, in vols. Deermine how long i will ake for a capacior in his ype of circui o discharge o a) half of is iniial charge b) 10% of is iniial charge CONNECTIONS Capaciors are used o sore and release elecric charges. They come in a variey of shapes, syles, and sizes and are used in a number of devices, such as surge proecors, audio amplifiers, and compuer elecronics. Resisors dissipae energy, ofen in a useful form such as hea or ligh. They also come in a variey of forms. 5.2 The Naural Logarihm MHR 265

8 12. Use Technology A pizza is removed from he oven a 0 min a a emperaure of 200 C. The emperaure, T, measured a he end of each minue for he nex 10 min is given in he able. Time (min) Temperaure ( C) a) Using exponenial regression, deermine a value of k so ha T() 200 e k models he emperaure as a funcion of ime. b) Show ha your funcion correcly predics he emperaure a 10 min. c) Predic he emperaure a 15 min and also afer a long period of ime. 13. a) Evaluae, using a calculaor, ln 2 ln 3. b) Evaluae ln 6. Compare hese resuls. c) Wha law of logarihms does his seem o verify? Recall Reasoning and Proving ha ln 2 log e 2. Represening Selecing Tools Rewrie his law of Problem Solving logarihms using Connecing Reflecing naural logarihms. Communicaing 1. Carbon-1 (C-1) is a radioacive subsance wih a half-life of approximaely 5700 years. Carbon daing is a mehod used o deermine he age of ancien fossilized organisms by comparing he raio of he amoun of radioacive C-1 o sable carbon-12 (C-12) in he sample o he curren raio in he amosphere, according o he equaion (ln 2 ) N () N e , where N() is he raio of C-1 o C-12 a he ime he organism died, N 0 is he raio of C-1 o C-12 currenly in he amosphere, and is he age of he fossil, in years. a) Calculae he approximae age of a fossilized sample found o have a C-1 : C-12 raio of i) 10% of oday s level ii) 1% of oday s level iii) half of oday s level b) Do you need he equaion o find all of he resuls in par a)? Explain your reasoning. c) Rearrange he formula o express i in erms of (isolae on one side of he equaion). C Exend and Challenge 15. Use Technology If you sudy daa managemen or saisics, you will learn abou he normal disribuion curve. Universiy exam scores ofen follow a normal disribuion. The normal disribuion curve is also someimes called he bell curve. a) Graph he funcion y e x2. b) Describe he shape of he graph. c) Wha is he maximum value of his funcion, and where does i occur? Exam Scores d) Esimae he oal area beween he curve and he x-axis. Use a rapezoid o approximae he curve. e) Esimae he fracion of his area ha occurs beween x 1 and x Mah Cones e log e2 x is equal o A 2x B x C x 2 D x E ln x Mah Cones If log x (e a ) log a e, where a 1 is a posiive consan, hen x equals A a B 1 a C a a D a a E a a MHR Calculus and Vecors Chaper 5