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1 Guide to Logrithms d Epoets Pul A. Jrgowsky, Rutgers-Cmde. Review of the Algebr of Epoets. Before discussig rithms, it is importt to remid ourselves bout the lgebr of epoets, lso kow s powers. Epoets re compct ottio to epress multiplictio of umber or vrible by itself: 3 9 Epoets with egtive sigs (-) re defied to me the vrible is i the deomitor, rised to : Importt fct #: Whe two powers of the sme uderlyig vrible re multiplied, the epoets dd. Emie the followig to see why: I geerl: c d c d Revised 7/3/0. Plese report y typos, errors, or commets to pul.jrgowsky@rutgers.edu

2 Importt fct #: Whe power of is divided by power of, the epoets must be subtrcted (umertor epoet deomitor epoet), becuse some of the s ccel out d the swer is wht is left over: I geerl: 5 5 c d cd There is iterestig implictio of this property. Wht is rised to the zero power? My hve ituitio tht it should be zero, but i fct ythig to the zero power is. Do t believe it? Look t this: 0 Epoets re just multiplictio, d multiplictio pivots roud, ot zero. Whe I sy it pivots roud, I me tht umber times its iverse is, ot 0. (For dditio the pivot poit is zero, so tht plus its dditive iverse, -, equls 0.) Thus, power is. c lso be writte 0. Agi, y umber to the zero Importt fct #3: Whe power of vrible is itself rised to power, the epoets must be multiplied, s show below: I geerl: 3 3 c d cd

3 . Defiitio of Logrithms. With ll the prelimiries out of the wy we c filly tlk bout rithms. Look t the followig word equtio. epoet bse vlue I other words, we rise some umber clled the bse to power d we get vlue. For emple,? Well, it s ot hrd to see tht the swer is 6. But suppose we kew? 6 the swer, but ot the epoet:. I this cse, we re lookig for the power to which the umber hs to be rised to get 6. The swer is. Logrithms re just epoets. All rithms re swers to questios i this form: to wht power must the bse be rised to get the give umber. Here is how we write the? sme questio i two differet wys: 6? They both sy the 6, or sme thig: to wht power does hve to be rised to get 6? So, to repet, rithm is just epoet. Here re some emples: 6 becuse 6 6 becuse becuse ,000 becuse 0 0, becuse 5 5 The lst oe seems little poitless, but I ll come bck to it. Logrithms re fuctios, like f (), so you relly should write them i fuctio ottio 6. This does ot me times 6, but the with pretheses s bove: output of the fuctio for the iput of 6. Most of the time, however, people do t bother writig the pretheses uless they re ecessry to specify the order or opertios or to mke the equtio look prettier. I will geerlly omit them if the iput to the fuctio is simple, like or y, but iclude them if the rgumet is more comple, just to be cler o wht is beig ged. Bses. Ay positive rel umber c be the bse of rithms. My differet bses re used, especilly i the scieces, but most of the time you re oly goig to see just two bses: 0, d e =

4 While e my seem like strge umber to use s bse for ythig, it turs out tht rithms with this bse hve ice properties tht mke them esy to work with t higher levels of mthemtics, e.g. clculus. Just thik of e s costt, like π = Both re o-repetig ifiite decimls, so it is coveiet to use symbol. If you eed to perform clcultio, you hve to roud it off. Logrithms bsed o 0 re clled commo rithms, while rithms bsed o e re clled turl rithms. They re usully writte s follows: 0 l e So if you see with o bse specified, ssume the bse is 0. If you see l, the bse is the umber e. However, be wre tht some uthors use whe they me l. Iverse Fuctios. Tkig rithms of bse b d risig b to some power re iverse fuctios: b b 0 0 l 0 0 l b b e e These idetities just follow from the defiitio of rithms. Wht power does b eed to be rised to i order to get b? Well, just. Tke other look t the rithm, bse 5, of 5 bove, the oe tht I sid I would get bck to. Logrithms s Reltive Chges. For smll chges, the chge i l() times 00 is pproimtely the percetge chge i. For emple, 5 percet icrese i results i lmost 0.05 icrese i l(), regrdless of the strtig vlue of. Emple: l() Chge 5 (5% icrese) 0.088, , Chge 67.5 (5% icrese) Clculus, if you kow it, shows why: d l d d l l for smll chges d

5 Thus, if you hve iterest i reltive chges (chge i percetge terms) rther th bsolute chges, your lysis should focus o l() s the depedet vrible. 3. Useful Properties of Logrithms. The lst sectio eplied wht rithms re. This sectio eplis why you should cre. There re some properties of rithms tht tur out to be very useful for simplifyig d mipultig equtios. I will epress these properties i terms of turl rithms (l = e), but they work ectly the sme for rithms to y bse. To epress these properties for rithms to other bse b, just replce l with b d e with b i the formuls below. To epress them for commo rithms, replce l with 0 (or just ) d e with 0. Useful Property #: First, rithms chge multiplictio ito dditio: l y l l y Why? It s just cosequece of the rule for multiplyig powers described bove. Let s defie two umbers, m d tht re the turl rithms of d y, respectively. m m l e l y e y Now let s look t the product of d y: m m y e e e Now we tke the turl rithm of both sides of the equtio, preservig the equlity: l y l e l l y m m The chgig of multiplictio to dditio is the bsis of slide rules. A slide rule hs two rithmic scles; you slide oe of the scles reltive to the other so tht you physiclly dd up the two rithms d red off the result. A lost rt. 5

6 Useful Property #: Secod, rithms chge divisio ito subtrctio. l l l y y The proof is quite similr. Usig the sme defiitio of m d s bove: l e y e m l e y m l l y e m m Useful Property #3: Third, rithms chge epoets ito multiplictio. l l To see this, we use the defiitio of m s the turl rithm of from bove: m m e e This result follows from Importt Fct #3 bove. Now we both sides: l m l e m l Thigs you c t do. It is temptig sometimes to try to do thigs with rithms tht you just c t do. For emple: l y l l y l y l l y l l l l l y y l l y l y Stick to the three properties described bove d you wo t slip ito y of these errors. 6

7 Covertig Bses. If you eed to covert rithms from oe bse to other, it is pretty esy. Here is the rule to covert rithm from bse to bse b: b b For emple, if we kow tht the turl (l = e) of 00 is.6057, but we eed the bse 0 rithm, we divide by the turl of 0. l l We probbly could hve figured tht out without the formul! But it illustrtes the poit. Here is where the bse coversio formul comes from: b b b b b b Sice the deomitor is costt, you c see tht s to oe bse re strictly proportiol to s i y other bse.. Applictios. Lierizig fuctio. How c we use lier regressio o o-lier fuctios? Q AL K l Q l A l L l K Let la be the costt, dd disturbce term, d you c regress t will. Problems with Iterest Rtes. How c we solve for the iterest rte, r? For emple, wht is the rte of retur if $0,000 ivestmet icreses to 8,000 i 5 yers? Strtig with the bsic compoud iterest formul, we c derive r. 7

8 0 l X X X r l X l X l r l X l( r) r e r e l X 0 l X0 l X l X0 0 I c ever remember this formul, d so I hve to re-derive it every time I eed it. Good thig I kow my rithms. To swer the questio posed bove: l X l X0 l8000 l % r e e e Try solvig for, d use tht formul to see how my yers it would tke to double your moey for give iterest rte. 5. Summry. The tble below summrizes the iformtio bove, d shows the dulity of the lgebr of epoets d the properties of rithms. m Defiitios e, e y m l, l y, b,, y ll positive rel umbers, e Epoets e 0 Logrithms l 0 e e e l l e e l e, e m m Multiplictio y b b b ly l l y Divisio m e m e l l l y y e y m m Powers e e l l b b Chge of Bse b b 8