Logarithms and Exponential Functions Handout

Transcription

1 Logarihms and Exponenial Funcions Handou Logarihms were invened by John Napier o eliminae he edious calculaions involved in muliplying, dividing, and aking powers and roos of large numbers ha occur in asronomy and oher sciences. However, logarihms arise in mos echnology, science, and engineering siuaions. Problems associaed wih exponenial growh and decay are very common for bioengineers and bio-echnicians because logarihms are inverses of exponenial funcions. Logarihms also urn ou o be useful in he measuremen of he loudness of sounds, he inensiy of earhquakes, and many oher phenomena of immediae ineres o echnical people. Exponenial Growh and Decay Like many opics ha echnicians, scieniss, and engineers deal wih, logarihms are acually mahemaical operaions and are usually associaed wih an equaion. Technicians undersand ha such equaions are useful and help people from making misakes. One example is he producion of drugs from baceria. No all baceria are harmful o humans. Bio-engineers and echnicians ofen creae liquid environmens where baceria can grow and muliply. Afer a prescribed ime period, he baceria will have produced a specific amoun of a proein ha can hen be removed from he liquid and made ino a drug o help preven or even cure specific human diseases. One of he many challenges he echnician has o deal wih is o mainain a specific populaion (number) of baceria in he liquid even hrough hey are growing, muliplying, and dying all he ime. One ool ha he echnician uses is an equaion ha predics he number (amoun) of baceria ha will be in he liquid a some specific ime afer he iniial seed populaion was placed in he liquid. Mahemaicians usually describe such an exponenial growh equaion in he following way. Le y be he amoun or number presen a he beginning of his growh process, a ime usually defined as =. For baceria, his is usually a large number well over a one million. Then, under cerain condiions, he amoun presen a any ime (a value greaer han zero) is λ given by he exponenial funcion, y = ye, where he Greek leer lambda, λ, is he usual symbol ha is used o represen he rae of change. (Lambda is a popular Greek leer in engineering and science. I is also he usual symbol o denoe a wavelengh.) When λ >, he funcion describes growh (e.g. populaion growh). When λ <, he funcion describes decay (e.g. radioacive decay). For example, if lambda were equal o + baceria per second, he number of baceria, i.e. baceria populaion, would increase by baceria every second. If lambda were equal o - 4 baceria/second, he baceria populaion would decrease by 4 baceria every second. I may be easier o see how he equaion for exponenial growh or exponenial decay works if an acual example wih numbers is used. However, o make sure ha you undersand ha echnicians ofen change he leers and symbols used as variables in an exponenial equaion, we will change he leers for he variables in his example. For he people populaion example discussed in he nex paragraph, he symbol n will be used o indicae he iniial number of people, insead of he symbol, y, ha was used in he previous paragraph. Page of 8 4 High School Technology Iniiaive (HSTI) Educaional Maerials: The ATOM: Applied Mah

2 Populaion Growh The growh in he populaion of he world is expeced o follow an exponenial equaion. If n is he iniial size of a populaion experiencing exponenial growh, hen he number of people, he populaion, a any ime is idenified by he variable symbol n (). For example, if he echnician used he symbol n(5), i would represen he a specific number people ha exis afer 5 unis of ime. The change in populaion afer ha 5 unis of ime would simply be he number of people afer 5 unis of ime, n() when =5, minus he number of people o sar wih, n. λ Using hese defined variable symbols, he exponenial equaion becomes n( ) = ne, where λ is he relaive rae of growh expressed as a fracion of he populaion. This equaion is very easy o use. The n () symbol is jus he symbol for he answer. The n symbol represens he number of people o sar wih. The e λ erm is jus he number.783 raised o he power deermined by he muliplicaion of he lambda value and he ime of ineres, λ. Example of equaion seup The populaion of he world in was 6. billion people, and he esimaed relaive populaion growh was.4% per year. Find a funcion ha models his growh. Soluion Unless oher specific informaion is known, mos people will use he exponenial populaion growh equaion (model) wih n = 6. x 9 people (6. billion people) and λ =. 4, (.4% / =.4). Wih his in mind, all ha needs o be done is o subsiue λ he proper numbers ino he proper place in he general equaion, n( ) = n e. Example Problem The growh of he populaion of he world is modeled as people will be on he earh in 5? n.4 ( ) = 6.e. How many Soluion Please noe:. The ime value for "" in he equaion is he amoun of ime ha is being modeled and is equal o 5 years, or, The lambda value, λ, is he rae of change expressed no as a percen bu as a decimal number. Since.4% per year is he same as.4 per year, λ = The saring populaion, n, is equal o 6. billion, (6. x 9 ) people. 4. Wih his informaion available he equaion can be wrien wih he appropriae 9.4 subsiuions in mind o ge o he specific exponenial funcion n () = 6. x e and he specific equaion: n(5) = 6 x 9 e (.4)(5). 5. This equaion can go hrough he following simplificaions n(5) = 6 x 9 e (.7), and 6. n(5) = 6 x 9 (.75 ) = (6)(.75) x 9 n(5) = x 9 ANSWER There will be rillion people in he world in 5. Page of 8 4 High School Technology Iniiaive (HSTI) Educaional Maerials: The ATOM: Applied Mah

3 Radioacive Decay Half-life Radioacive subsances decay by sponaneously emiing radiaion. The rae of decay is direcly proporional o he mass of he subsance. This is also an exponenial funcion and i is analogous o populaion growh, excep ha he mass of radioacive maerial decreases. The λ model, m( ) m e, ha describes his decay is very similar o he exponenial growh model, = λ ( ) = ne, used in he populaion problem in he las secion. n Again, don' be fooled by he change in symbols, bu compare he number of symbols and heir arrangemens. Everyhing is he same excep he sign in he exponen erm. The m() symbol serves he same role as he n() above. The m has he same meaning as he n in he las example and he lambda symbol, λ, sill represens a rae of change value, only his ime λ is he rae of decay expressed as a proporion of he mass when m is he iniial mass. A echnician on a US Navy nuclear powered aircraf carrier emphasizes specific ime values when describing he decay of a radioacive maerial. The half-life, ½, of a maerial is he ime required for half he maerial o decay. Thus, in one half-life he amoun of maerial ha remains is / of he saring amoun and afer wo half-life periods, only /4 ( ½ x ½ ) of he original amoun of maerial will be lef. Using a mahemaics rick abou solving exponenial equaion learned in school, he echnician can develop a formula for deermining he half-life of a radioacive maerial if he lambda, λ, for ha maerial is already known. λ As saed above, he model for exponenial decay is m( ) = m e. The rick is o undersand ha when = he is subsiued ino he model and he following algebra can be done because e λ also equals e and any number raised o he zero power equals. Tha is o say ha e =. Thus: λ m( ) = m e becomes m() = me λ or m() = m () Or in a verbal forma, since m is he amoun of radioacive maerial when =, he amoun of maerial afer one half life will be ( ½)( m ). This very simple idea ha if you sar wih m amoun of radioacive maerial and wai one half life, you will only have / of ha saring amoun lef, hus can be reused in he general λ model equaion, m( ) m e. = λ = e And, remembering ha / can also be yped as -, you can ake naural logarihms, he ln, of each side: λ ln = lne noe ha ( λ = λ, and e = e ) Since he equaion is now in is logarihmic form, he exponens (-) and (- λ ( ) ) can be placed in fron of he ln erms as coefficiens in he equaion saed below as: Page 3 of 8 4 High School Technology Iniiaive (HSTI) Educaional Maerials: The ATOM: Applied Mah

4 ln = λ lne Now i is really simple o solve for if we remember ha he naural logarihm of he number " e" is equal o he number. Tha is o say ha ln e = and he final soluion for This formula, = = ln λ ln λ becomes:, allows he echnician o find he half-life if we know he rae of decay, λ, for any maerial(s) ha migh be involved in a nuclear reacor. These half-lives of radioacive elemens vary from very long o very shor. Here are some examples. Elemen Thorium-3 Uranium-35 Thorium-3 Pluonium-39 Carbon-4 Radium-6 Cesium-37 Sronium-9 Polonium- Thonium-34 Iodine-35 Radon- Lead- Krypon-9 Half-life 4.5 billion years 4.5 billion years 8, years 4,36 years 5,73 years,6 years 3 years 8 years 4 days 5 days 8 days 3.8 days 3.6 minues seconds Example Plo he decay of he following radioisoopes (a) Sronium-9 and (b) Molybdenum-99. Page 4 of 8 4 High School Technology Iniiaive (HSTI) Educaional Maerials: The ATOM: Applied Mah

5 Soluion In general, if m is he iniial mass of a radioacive subsance wih half-life mass remaining a ime is modeled by he funcion m λ ( ) = m e, where, hen he λ = ln. Wih his knowledge in hand, he decay curve can be developed by deermining he m() values when he ime equals half-life period, half-life periods, ec. (a) Sronium-9 Radioacive Decay of Sronium-9 Mass of Sronium Aoms, grams half-life half-life half-life half-life half-life (8.7 yrs) Time (years) Page 5 of 8 4 High School Technology Iniiaive (HSTI) Educaional Maerials: The ATOM: Applied Mah

6 (b) Molybdenum-99 Radioacive Decay of Molybdenum-99 Number of Aoms half-life half-life half-life half-life hafl-life (68.7 yrs) Time (years) Example Techneium-99 is used o form images of inernal organs in he body and is ofen used o 99 deermine hear damage. This nuclide, Tc, decays o ground sae by gamma emission. The rae of decay is.6 x - d/hr. Wha is he half-life of his nuclide? Soluion Known: λ =.6 Unknown: Equaion: = ln λ d / hr ln Solve: = 6hr ANSWER: The half-life of Tc-99 is approximaely 6 hrs..6 Example 3 (ouside class exercise) In 89 Carl Auervon Welsbach (858-99) developed a horium manle for gas lanerns. The manle of a lanern is he whie looking mesh ne maerial ha glows when burning gas is passed hrough i. Today we use manles in Colman camping lanerns and emergency lighs ha run on propane. Recenly Colman Co. phased ou he use of horium manles and replaced hem wih yrium manles. These new manles do no glow as brighly as he old horium manles, bu yrium is no radioacive and is, herefore, safer o use. Assuming his horium manle had he same mass as he new replacemen yrium manle, how much manle maerial was radioacively decayed in half-life of a horium manle? Page 6 of 8 4 High School Technology Iniiaive (HSTI) Educaional Maerials: The ATOM: Applied Mah

7 Soluion This is an easy mahemaics problem, bu you will have o go o a spors sore o find ou how much a yrium manle weighs. You can find manles in heir own packages near where he Colman lanerns are sold. Once you ge ha value, jus divide ha mass in half and you will know how much of he manle will be around afer half-life. Good luck huning a he spors sore! Logarihmic Scales As suggesed a he beginning of his handou, here are many echnology relaed applicaions for logarihms. Anyime a physical quaniy varies over a very large range, i is ofen convenien o ake is logarihm in order o have a more manageable se of numbers. Once his is done, a echnician, engineer, or scienis will develop a scale ha helps pu he value of he logarihm ino perspecive wih he physical sysem being sudied. A few examples presened below include he ph scale, which measures acidiy; he Richer scale, which measures he inensiy of earhquakes; and he decibel scale, which measures he loudness of sounds. ph Scale Chemiss measure he acidiy of a soluion using a formula proposed by Sorensen in He defined ph = log [ H ], where [ H ] is he concenraion of hydrogen ions measured in moles per lier or (M). He did his o avoid very small numbers and negaive exponens. For example, if [ H ] 4 M + =, hen 4 ph = log( ) = ( 4) = 4. Soluions wih a ph of 7 are defined as neural, hose wih ph < 7 are acidic, and hose wih ph > + H decreases by a facor of. 7 are basic. Noice ha when he ph increases by one uni, [ ] The Richer Scale In 935 he American geologis Charles Richer defined he magniude M of an earhquake I o be M = log, where I is he inensiy of he earhquake (measured by he ampliude of a S seismograph reading aken km from he epicener of he earhquake) and S is he inensiy of 4 a sandard earhquake (whose ampliude is micron = cm). S For example, he magniude of a sandard earhquake is M = log = log =. S The Decibel Scale According o he Weber-Fechner Law, he psychological sensaion of loudness varies wih he logarihm of he sound inensiy. The inensiy level β, measured in decibel (db), is defined I as β = log, where I is he inensiy of he source in was per square meer ( W / m ), and I I W / = m is he hreshold of hearing measured a a frequency of herz. Page 7 of 8 4 High School Technology Iniiaive (HSTI) Educaional Maerials: The ATOM: Applied Mah

8 For example, he inensiy level of he barely audible reference sound is I β = log = log = db. The able below liss decibel inensiy levels for some common I sounds. Source of Sound β (db) Source of Sound β (db) Je akeoff 4 Heavy raffic 8 Jackhammer 3 Ordinary raffic 7 Rock concer Normal conversaion 5 Pain Whisper 3 Subway Rusling leaves - Page 8 of 8 4 High School Technology Iniiaive (HSTI) Educaional Maerials: The ATOM: Applied Mah