The Natural Logarithm
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1 The Naural Logarihm The Power Rule says n = n + n+ + C provie ha n. The formula oes no apply o. An anierivaive F( of woul have o saisfy F( =. Bu he Funamenal Theorem implies ha if > 0, hen Thus, plays he role of F(. Define he naural log funcion ln by By consrucion, if > 0, ln = =. for > 0. ln =, an = ln + C. represens he area uner f( = y from o : f( = / ln Bu why is his calle a logarihm? You ve probably seen logarihms use like his: log 8 = 3 because 3 = 8.
2 So I s no clear wha ln = has o o wih raising numbers o powers. Well, for one hing, ln has many properies you epec a logarihm o have. For eample, ln = = 0. You epec he log of a prouc o equal he sum of he logs. If a an b are posiive numbers, hen ln(ab = ab = a + In he secon inegral, le u =, so u =, an = a u. When = a, u = ; when = ab, u = b. a a a ab + a a = b + u = lna + lnb. u In oher wors, ln(ab = lna + lnb. In similar fashion, you can verify ha ab a. ln a b = lna lnb an lnr = r ln. Thus, here is some jusificaion in calling ln a logarihm, because i has he same properies you epec logs o have. I urns ou ha he whole sory is backwars! When you iscuss logs as he opposie of powers, you are acually being a lile sloppy. To efine he familiar logs (an eponenials wih mahemaical precision, wha you acually o is o efine ln an is inverse e firs, as I ve one above. Then you efine he oher logs an powers using ln an e. For eample, if a an b are posiive numbers, efine log a b = lnb lna. I s possible o check ha logs base a, as I ve jus efine hem, behave he way you epec logs o behave. Here are some aiional properies of ln. Firs, ln = > 0 for > 0. Therefore, he graph of ln is increasing for > 0. Moreover, ln = < 0 for > 0. Therefore, he graph of ln is concave own for > 0. Ne, consier he following picure: f( = / 3 4
3 The area uner he curve from o 4 is ln4. I is greaer han he sum of he areas of he hree recangles, so ln 4 > = 3 >. If n is a posiive ineger, hen So if > 4 n, hen n ln 4 > n, or ln 4 n > n. ln > ln4 n > n. Since n is an arbirary posiive ineger, I can make ln arbirarily large by making sufficienly large. This proves ha lim ln = +. + Here s he graph of ln: y y = ln Eample. The iffereniaion formula for ln works ogeher wih he oher iffereniaion rules in he usual ways. ln( = ln(sin + cos = sin +. 3 [ (ln 7 + ln( 7 ] = [ (ln ln ] ( = 7(ln 6 (ln( ( ln = (ln ( ln (ln (ln + = ( ln(ln + ln = (ln. ( ( ln If I say ha f( = g( he erivaive of f( is g( hen g( shoul be efine wherever f( is efine. Therefore, i is no really correc o say wihou he qualificaion > 0 ha ln =. For is efine for 0, whereas ln is only efine for > 0. 3
4 I urns ou ha he correc saemen is: ln = for 0. For > 0, his is he same as he ol formula. For < 0, =, so ln = ln( = =. So he upae anierivaive formula is = ln + C. You can omi he absolue value signs if he quaniy insie is never negaive. For eample, i urns ou ha + = ln( + + C. (Deriving his formula requires an inegraion echnique calle subsiuion. However, you can check ha i s correc by iffereniaing ln( + o ge. I can omi he absolue values aroun he + +, because + is always posiive. Eample. 5 = ln 5 + C. an = ln cos + C = ln sec + C. 3 + = 3 ln C. Eample. You can use logarihmic iffereniaion o compue erivaives which are ifficul o compue in oher ways. For eample, suppose you wan o iffereniae y =. You can use he Power Rule, because he eponen is, no a number. You can use he rule a = a lna (which you ll see laer, because he base is, no a number. Insea, ake logs of boh sies: lny = ln = ln. Differeniae implicily: Hence, y y = + ln. y = y ( + ln = ( + ln. By a similar proceure, you can iffereniae complicae proucs an quoiens. For eample, o ( iffereniae y =, ake logs of boh sies: 5 + 3(sin 4 lny = ln ( (sin 4 = ln( ln ( 5 + 3(sin 4 = ln( ln ln(sin 4 = 4
5 0 ln( + 3 ln( ln sin. To simplify, I use (in orer he following properies of logs: Differeniae implicily: Hence, ( 0 y = y + 3 ln a b = lna lnb, lnab = lna + lnb, lnar = r lna. y y = cos = sin cos sin. ( ( (sin ( cos. sin Eample. Compue ( + 4. Le y = ( + 4. Taking logs an bringing he power own, I ge lny = ln( + 4 = 4 ln( +. Differeniae boh sies, using he Chain Rule on he lef an he Prouc Rule (an Chain Rule on he righ: y ( y = (4 + [ln( + ]( Muliply boh sies by y o clear he fracion on he lef, hen subsiue y = ( + 4 : ( ( ( ( y = y ( 4 + [ln( + ](4 3 = ( + 4 ( 4 + [ln( + ]( c 006 by Bruce Ikenaga 5
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