Chapter 4 Logarithmic Functions
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1 4.1 Logarihms and Their Properies Chaper 4 Logarihmic Funcions Wha is a Logarihm? We define he common logarihm funcion, or simply he log funcion, wrien log 10 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. Logarihms are Exponens Logarihms are jus exponens! 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, 10 log N = N 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.
2 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, ln x is he power of e ha gives x or, in symbols, ln x = y means e y = x, and y is called he naural logarihm of x. The funcions e x and ln x are inverses. Properies of he Naural Logarihm By definiion, y = ln x means x = e y. In paricular, ln1 = 0 and lne = 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, ln( ab) ln a ln b a ln ln a ln b b ln( b ) ln b. 4.2 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. 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. 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. Convering Beween Q = ab and Q = ae k Any exponenial funcion can be wrien in eiher of he wo forms: Q = ab or Q = ae k If b = e k, so k = lnb, he wo formulas represen he same funcion.
3 4.3 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 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 or lim f ( x) a x x The graph of f has a verical asympoe of x = a if lim f ( x) or lim f ( x) or lim xa xa xa or boh. 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. Graph of Naural Logarihm The naural log and he common log have similar graphs.
4 Chaper 3 & 4 - Review Examples 1. Wha is he half-life of a radioacive subsance ha decays a 10.4% per minue? 2. Caffeine leaves he body a a coninuous rae of 17% per hour. How much caffeine is lef in he body 4 hours afer drinking a cup of coffee conaining 100 mg of caffeine? 3. Find a formula for he funcion f wih f and a. linear b. exponenial 4. Find a formula for he exponenial funcion if V h iniially worh $10,000 ha loses half is value every 5 years. f 2 12 assuming i is: gives he value of an iem 5. A populaion doubles in size every 15 years. Assuming exponenial growh, find he: a. Annual growh rae b. Coninuous growh rae 6. Wha is he domain and range of g( x) ln x? 7. Wha is he domain and range of g( x) log x? 8. A own has populaion 3000 people a year =0. Wrie a formula for he populaion, P, in year if he own: a. Grows by 200 people per year. b. Grows by 6% per year. c. Grows a a coninuous rae of 6% per year. d. Shrinks by 50 people per year. e. Shrinks by 4% per year. f. Shrinks a a coninuous rae of 4% per year. 9. Wrie he exponenial funcion Q 240(1.0376) in he form accurae o hree decimal places. Q ae k. Find k Wrie he exponenial funcion y 75e in he form y ab. Find b accurae o four decimal places. If is measured in years, give he percen annual growh or decay rae and he coninuous growh or decay rae per year.
5 11. In July 2005, he Inerne was linked by a global nework of abou million hos compuers. The number of hos compuers has been growing approximaely exponenially and was abou 36.7 million in July a. Find a formula for he number, N, of he inerne hos compuers as an exponenial funcion of, he number of years since July 1998, using he form N b. A wha coninuous annual percen rae does N increase? c. If his growh rae coninues, how many years will i ake for he number of hos compuers o double? d. Wha would he annual growh rae be? k ae. 12. Solve he following equaions exacly if possible. 3log 5 6 a. ln xln x1 b. 1 2 c. 2ln 3x 5 8 d e. 7e 2e f. x e 3x 5 g. 2 ln x x 13. In 1980, he populaion of Mexico was million and increased a a consan annual percen rae of 2.6 % during he early 1980 s. When will he populaion be 76.6 million? Wha does he formula predic when 0 and 5. Wha does hese values ell you abou he populaion of Mexico?
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