5.1 - Logarithms and Their Properties
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1 Chaper 5 Logarihmic Funcions Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We wan o solve he following equaion for : so we used a graphical mehod o approximae. 3 4 Since and 10 10,000 so ,000, is beween 3 and 4, bu how do find exacly? We can use he log funcion. Wha is a Logarihm? We define he common logarihm funcion, or simply he log funcion, wrien log10 x or log x, as follows: If x is a posiive number, In oher words, if log x is he exponen of 10 ha gives x. y log x hen 10 y x. Example 1 Rewrie he following saemens using exponens insead of log. a. log b. log c. log Example 2 Rewrie he following saemens using logs insead of exponens. 5 a ,000 b c Logarihms are Exponens Logarihms are jus exponens! Thinking in erms of exponens is ofen a good way o answer a logarihm problem. 1
2 Chaper 5 Logarihmic Funcions Example 3 Wihou a calculaor, evaluae he following, if possible: a. log 1 b. log 10 c. log 1,000,000 d. log e. log 1 10 f. log 100 Logarihmic and Exponenial Funcions are Inverses The operaion of aking a logarihm undoes he exponenial funcion; he logarihm and he exponenial funcions are inverse funcions. In paricular: For any N, log 10 N N and for N 0, log 10 N N Example 4 Evaluae wihou a calculaor. 8.5 a. log 10 b. log c. 10 log x3 2
3 Chaper 5 Logarihmic Funcions Properies of Logarihms Properies of he Common Logarihm By definiion, y = log x means 10 y = x. In paricular, log1 = 0 and log10 = 1. The funcions 10 x and log x are inverses, so hey undo each oher: log(10 x ) = x for all x, 10 log x = x for x > 0. For a and b boh posiive and any value of, log( ab) log a log b a log log a log b b log( b ) log b. Example 5 Solve ,000,000 for. Example (Exercise #18 on pg. 186) Solve for. 7 3
4 Chaper 5 Logarihmic Funcions The Naural Logarihm When e is used as he base for exponenial funcions, compuaions are easier wih he use of anoher logarihm funcion, called log base e. For x 0, or, in symbols, ln x is he power of e ha gives x ln x y means y e x, and y is called he naural logarihm of x. Jus as he funcions 10 x and log x are inverses, so are he funcions Properies of he Naural Logarihm By definiion, y = ln x means x = e y. In paricular, ln1 0 and ln e 1. The funcions e x and ln x are inverses, so hey undo each oher: ln(e x ) = x for all x, e ln x = x for x > 0. For a and b boh posiive and any value of, x e and ln x. ln( ab) ln a ln b a ln ln a ln b b ln( b ) ln b. Example 6 Solve for x: 2x a. 5e 50 b. 3 x 100 4
5 Chaper 5 Logarihmic Funcions Example (No an Example in he Secion) Solve he following equaions exacly if possible for x or : a. 17e 18e b. 5 3log 6 ln x ln x1 c ln 3x 5 8 d. x e. e 3x 5 f. ln x x 2 5
6 Chaper 5 Logarihmic Funcions Logarihms and Exponenial Models The log funcion is ofen useful when answering quesions abou exponenial models. Because logarihms undo he exponenial funcions, we use hem o solve many exponenial equaions. Example 1 In Example 3 on pg. 151, (A 200 ug sample of carbon-14 decays according o he formula Q , where is in housands of years. Esimae when here is 25g graphically. Now solve using logarihms. of carbon-14 lef) we solved he equaion Example 2 The US populaion, P, in millions, is currenly growing according o he formula years since When is he populaion prediced o reach 350 million? P 299e 0.009, where is in Example 3 The populaion of Ciy A begins wih 50,000 people and grows a 3.5% per year. The populaion of Ciy B begins wih a larger populaion of 250,000 people bu grows a he slower rae of 1.6% per year. Assuming ha hese growh raes hold consan, will he populaion of Ciy A ever cach up o he populaion of Ciy B? If so, when? Doubling Time Evenually, any exponenially growing quaniy doubles, or increases by 100%. Since is percen growh rae is consan, he ime i akes for he quaniy o grow by 100% is also a consan. This ime period is called he doubling ime. Example 4 a. Find he ime needed for he urle populaion described by he formula P o double is iniial value. b. How long does his populaion ake o quadruple is iniial size? 6
7 Chaper 5 Logarihmic Funcions Example 5 A populaion doubles in size every 20 years. Wha is is coninuous growh rae? Half-Life An exponenially decaying quaniy decreases by a facor of 2 in a fixed amoun of ime, called he half-life of he quaniy. Example 7 Carbon-14 decays radioacively a a consan annual rae of %. Show ha he half-life of carbon-14 is abou 5728 years. Convering Beween Q = ab and Q = ae k Any exponenial funcion can be wrien in eiher of he wo forms: k Q ab or Q ae k If b e, so k ln b, he wo formulas represen he same funcion. Example 8 k The quaniy, of a subsance decays according o he formula Q Q0e, where is in minues. The half-life of he subsance is 11 minues. Wha is he value of k? Example 9 Conver he exponenial funcion P o he form k P ae. Example 10 Conver he formula Q 0.3 7e o he form Q ab. 7
8 Chaper 5 Logarihmic Funcions Example 11 Assuming is in years, find he coninuous and annual percen growh raes in Examples 9 and 10. Example 12 Find he coninuous percen growh rae of Q , where is in housands of years. Example 13 Wih in years, he populaion of a counry (in millions) is given by P 21.02, while he food supply (in millions of people ha can be fed) is given by N Deermine he year in which he counry firs experiences food shorages. Review Example 6 8
9 Chaper 5 Logarihmic Funcions The Logarihmic Funcion The Graph, Domain, and Range of he Common Logarihm The domain of log x is all posiive numbers. The range of log x is all real numbers. The log funcion is increasing and is graph is concave down, since is rae of change is decreasing. Graphs of he Inverse Funcions y = log x and y = 10 x Graph of Naural Logarihm The naural log and he common log have similar graphs. Example 1 Graph y ln x for 0 x 10. Asympoes Le y = f (x) be a funcion and le a be a finie number. The graph of f has a horizonal asympoe of y = a if lim f ( x) a x or lim f ( x) a x or boh. The graph of f has a verical asympoe of x = a if lim xa f ( x) or lim xa f ( x) or lim xa f ( x) or lim xa f ( x) Noice he process of finding a verical asympoe is differen from he process for finding a horizonal asympoe. Verical asympoes occur where he funcion values grow larger and larger, eiher posiively or negaively, as x approaches a finie value. Horizonal asympoes are deermined by wheher he funcion values approach a finie number as x akes on large posiive or large negaive values. Example (Exercise #8 on pg. 203) Graph he funcion and label all asympoes and inerceps. Sae he domain. yln x 1 Example (Exercise #12 on pg. 203) Wha is he value (if any) of he following? x a. e as x b. ln x as x 0 9
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