4.1 - Logarithms and Their Properties

Transcription

1 Chaper 4 Logarihmic Funcions Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number, In oher wds, if log x is he exponen of 10 ha gives x. y log x hen 10 y x. Example 1 on pg. 152 in Tex Rewrie he following saemens using exponens insead of log. a. log b. log c. log Example 2 on pg. 152 in Tex 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. 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: F any N, log 10 N N and f N 0, log 10 N N 1

2 Chaper 4 Logarihmic Funcions Example 4 on pg. 153 in Tex Evaluae wihou a calcula. 8.5 a. log 10 b. c. log log x3 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 f all x, 10 log x = x f x > 0. F 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 on pg. 154 in Tex Solve ,000,000 f. Example (Exercise #16 on pg. 157) Solve f. 7 2

3 Chaper 4 Logarihmic Funcions The Naural Logarihm When e is used as he base f exponenial funcions, compuaions are easier wih he use of anoher logarihm funcion, called log base e. F x > 0,, in symbols, ln x is he power of e ha gives x ln x = y means e y = 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 e x and ln x. 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 f all x, e ln x = x f x > 0. F 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. Example 6 on pg. 155 in Tex Solve f x: 2x a. 5e 50 b. 3 x 100 3

4 Chaper 4 Logarihmic Funcions Example (Exercise #46 on pg. 159) Solve f : e 18e Example (No an Example in he Secion) Solve he following equaions exacly if possible f x : 3log 5 6 a. ln x ln x1 b ln 3x 5 8 c. x d. e 3x 5 e. ln x x 2 4

5 Chaper 4 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 on pg. 159 in Tex In Example 3 on pg. 125, (A 200 ug sample of carbon-14 decays accding o he fmula Q , where is in housands of years. Esimae when here is 25g of carbon-14 lef) we solved he equaion graphically. Now solve logarihms using Example 2 on pg. 159 in Tex The US populaion, P, in millions, is currenly growing accding o he fmula P 299e, where is in years since When is he populaion prediced o reach 350 million? Review Example 3 on pg. 160 Doubling Time Evenually, any exponenially growing quaniy doubles, increases by 100%. Since is percen growh rae is consan, he ime i akes f he quaniy o grow by 100% is also a consan. This ime period is called he doubling ime. Example 4 on pg. 161 in Tex a. Find he ime needed f he urle populaion described by he fmula P o double is iniial value. b. How long does his populaion ake o quadruple is iniial size? Example 5 on pg. 161 in Tex A populaion doubles in size every 20 years. Wha is is coninuous growh rae? 5

6 Chaper 4 Logarihmic Funcions Half-Life An exponenially decaying quaniy decreases by a fac of 2 in a fixed amoun of ime, called he half-life of he quaniy. Example 6 on pg. 162 in Tex 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 fms: k Q ab Q ae If b k e, so k ln Example 8 on pg. 163 in Tex b, he wo fmulas represen he same funcion. Conver he exponenial funcion P o he fm annual percen growh raes, assuming is in years. P ae k. Find he coninuous and Example 9 on pg. 163 in Tex 0.3 Conver he fmula Q 7e o he fm growh raes, assuming is in years. Q ab. Find he coninuous and annual percen Example 11 on pg. 163 in Tex Find he coninuous percen growh rae of Q , where is in housands of years. 6

7 Chaper 4 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 Asympoes Le y = f (x) be a funcion and le a be a finie number. The graph of f has a hizonal asympoe of y = a if a x a x boh. The graph of f has a verical asympoe of x = a if xa xa xa xa Noice he process of finding a verical asympoe is differen from he process f finding a hizonal asympoe. Verical asympoes occur where he funcion values grow larger and larger, eiher posiively negaively, as x approaches a finie value. Hizonal asympoes are deermined by wheher he funcion values approach a finie number as x akes on large posiive large negaive values. Graph of Naural Logarihm The naural log and he common log have similar graphs. Example 11 on pg. 163 in Tex Graph y ln x f 0 x 10. Example (Exercise #10 on pg. 173) Graph he funcion and label all asympoes and inerceps. Sae he domain. yln x 1 7