Logarithmic Scales: the most common example of these are ph, sound and earthquake intensity.

Transcription

1 Numercy Itroductio to Logrithms Logrithms re commoly credited to Scottish mthemtici med Joh Npier who costructed tle of vlues tht llowed multiplictios to e performed y dditio of the vlues from the tle. Logrithms re used i my situtios such s: () () (c) Logrithmic Scles: the most commo emple of these re ph, soud d erthquke itesity. Logrithm Lws re used i Psychoy, Music d other fields of study I Mthemticl Modellig: Logrithms c e used to ssist i determiig the equtio etwee vriles. Logrithms were used y most high-school studets for clcultios prior to scietific clcultors eig used. This ivolved usig mthemticl tle ook cotiig rithms. Slide rules were lso used prior to the itroductio of scietific clcultors. The desig of this device ws sed o Logrithmic scle rther th lier scle. There is strog lik etwee umers writte i epoetil form d rithms, so efore strtig Logs, let s review some cocepts of epoets (Idices) d epoet rules. The Lguge of Epoets The power c e writte i epded form s:... [ for fctors] The power cosists of se d epoet (or ide). Bse The umer eig repeted. Epoet (or ide) The umer of times it is repeted. Emples: ^ Wé Multiplictio or divisio of powers with the sme se c e simplified usig the Product d Quotiet Rules. Cetre for Techig d Lerig Acdemic Prctice Acdemic Skills Digitl Resources Pge ctl@scu.edu.u [lst edited o 6 Novemer 07]

2 The Product Rule + m m Whe multiplyig two powers with the sme se, dd the epoets The Quotiet Rule m m Whe dividig two powers with the sme se, sutrct the epoets e e e e 6 6 The zero epoet rules c lso e used to simplify epoets. The Zero Epoet Rule 0 A power with zero epoet is equl to. The power rule c help simplify whe there is power to power. The Power Rule e ( ) 0 m ( ) m Whe power is rised to power, multiply the epoets. ( ) 8 66 ( ) 6 ( ) Two more useful Power Rules re: ( ) m m m or ( ) m m m or m m m m m m ( ) ( ) 6 6 The egtive epoet rule is useful whe power with egtive epoet eeds to e epressed with positive epoet. Pge

3 The Negtive Epoet Rule m or m m Tke the reciprocl d chge the sig of the epoet. m The frctio epoet rule estlishes the lik etwee frctiol epoets d roots. The Frctiol Epoet Rule m for uit frctios, or ( ) m or for y frctio m 8 8 p p Epoet Fuctios foud o Scietific Clcultor Fuctio Apperce of Key Emple Squre d d 9 Cue D D 8 Ay epoet for W ^ Wé Squre root 6 s or s s6 qd 7 Cue root S or S Geerlly qdgives S or S (lthough this vries etwee differet mkes d models) Pge

4 0qf q W00p.699 Ay root 0 00 F or Geerlly qfgives F or q W gives (lthough this vries etwee differet mkes d models) Pge

5 Numercy Defiitio of Logrithm Whe you thik out Logrithms you should thik out epoets (idices) The umer c e writte s. Q: Does tht me tht the Logrithm of is equl to? A: Prtilly! The rithm of does equl ut oly whe se of is used. As Logrithm, this c e writte s We kow tht 6 6 the Log (rithm) of 6 to the se 6 is The is the epoet (); the epoet is ecuse the se used ws is equivlet to writig is writte i epoetil form 6 6 is writte i rithmic form Cetre for Techig d Lerig Acdemic Prctice Acdemic Skills Digitl Resources Pge ctl@scu.edu.u [lst edited o 6 Novemer 07]

6 A few more emples: 6 is equivlet to writig is equivlet to writig Itegrl d frctiol epoets re lso possile is equivlet to writig 0 0 is equivlet to writig 8 (ltertively 0. ) 0. is equivlet to writig 0. Geerlisig ll of the ove: 7 0 is equivlet to writig 7 0 If umer (N) is writte s power with se () d epoet (e), such s e N the N e. The se,, must e positive umer ( > 0) d ot equl to ( ). The umer, N, must lso e positive (N > 0) Emples: Write the followig i rithmic form. Epoetil Form () Logrithmic Form 6 () (c) (d) 9 ( 9 is the sme s 9 ) 9 (e) 8 8 (f) c c Pge

7 The se is 6 The se is is equivlet to writig 6 6 The epoet is The of the umer is Note: The rithm of to y se is lwys 0. 0 Why? Rememer the zero epoet rule 0 0 writte s I Logrithmic Form ecomes 0 Video The Defiitio of Log(rithm) I the et set of questios, the rithmic form is give d is to e writte i epoetil form. Write the followig i epoetil form. Logrithmic Form () () Epoetil Form (c).. (d) 0 0 (e) 9 k 9 k Video The Defiitio of Log(rithm) Pge

8 Fidig Log Vlues Your scietific clcultor c fid the vlues of s. Most scietific clcultors hve two rithmic fuctios; Ordiry Logrithms d Nturl Logrithms. Ordiry Logrithms hve se of 0. Becuse we use se 0 umer system, it seems stright forwrd tht Logs with se of 0 re used. The Log key o scietific clcultor hs the pperce g. Whe you re clcultig the of umer, you c ssume it is with se of te uless it is idicted otherwise. I ordiry rithms, whe you fid the Log of umer, you re fidig the epoet whe se of 0 is used. Fid the vlue of Log (meig Log 0 ) O your clcultor, the sequece of keys is: g If Log this c e writte i epoetil form s 0. Fid the vlue of Log00 (meig Log 000) O your clcultor, the sequece of keys is: g If Log this c e writte i epoetil form s 000. Pge

9 Nturl Logrithms Nturl Logrithms hve se of e. You my lredy kow out π (pi) which is costt used i situtios ivolvig circles or cyclic evets. The costt e is used i situtios ivolvig growth d decy such s popultio growth. The costt e hs vlue.78 (rouded to deciml plces). The turl key o scietific clcultor hs the pperce h. The turl of umer c e writte s l N or e N. Whe you fid the turl of umer, you re fidig the epoet whe se of e (.78) is used. Fid the vlue of l (which is equivlet to ) e O your clcultor, the sequece of keys is:h.8878 The l.8878 which i epoetil form is.8878 e. Fid the vlue of l00 [or 00] e O your clcultor, the sequece of keys is:h The l which i epoetil form is e. Video Logs - Usig your clcultor It would e uusul to e sked to fid rithm of umer with se differet to 0 or e. If you were, the the sectio elow is relevt. Pge

10 Other Bses Fidig the vlue of rithm to se other th 0 or e c e performed o you clcultor. However, this requires the use of oe of Log Lws. The Log Lws re covered i other sectio. The Log Lw used here is clled the Chge of Bse Lw. This lw llows rithm with give se to e chged to ew se, the ew se eig oe tht is ville o your clcultor, tht is, se 0 or se e. The Chge of Bse Lw c e stted s: Log N LogN Log If se 0 is chose s the ew se, the the Lw c e writte s: Log0N LogN LogN Log Log If se e is chose s the ew se, the the Lw c e writte s: LogeN l N LogN Log l 0 e For emple: Fid the vlue of 0 Usig the Chge of Bse Lw: Usig the g key (to d.p.) Clcultor key sequece: g0pg Usig the h key. e 0 l (to d.p.) e l Clcultor key sequece: h0ph Pge 6

11 Workig i reverse! If you re preseted with questio where you kow the ordiry rithm of the umer d you wt to fid out wht the umer is, the you will e workig i reverse. Such questio will e preset like this oe: 0.77 Rememer, if o se is give the the implied se is 0. Writig i the se i the questio will ssist i gettig to solutio. The questio c e rewritte s: Now, usig the defiitio of rithm, Look closely t your clcultor, you will otice tht the SHIFT (or d F) of the key is G or (0 ). The d its iverse use the sme key! Usig the clcultor, for emple, qg0.77 The swer will e displyed. Rememer qg gives the G or (0 ) purposes the vlue of is. fuctio. For ll prcticl If the questio ivolves turl s, questio could e writte s: l. Rememer, l implies turl s, so the implied se is e, writig i the se i the questio will ssist i gettig to solutio. The questio c e rewritte s: e. Now, usig the defiitio of rithm, e. Look closely t your clcultor, you will otice tht the SHIFT (or d F) of the l key is H or ( e ). The l d its iverse use the sme key! Usig the clcultor, for emple, qh. The swer will e displyed. Rememer qh gives the H or ( e ) fuctio. For ll prcticl purposes the vlue of is 8.!. 8. e If the questio ivolves rithm of y se, questio could e writte s: This would e ulikely to occur ut if it did it is quite esily solved. Pge 7

12 Now, usig the defiitio of rithm, This time you will use fmilir key; the f or w. Usig the clcultor, for emple, f0.606 The swer.6907 will e displyed. For prcticl purposes the vlue of is.6 depedig o the ccurcy required Video Logs i reverse! Pge 8

13 Activity. Write the followig i rithmic form. () () 0 (c)..8 (d) (e) (f) (g) (h) (i) 7.89 e (j) e. Write the followig i epoetil form. () 00 () 0 0 (c).6 0. (d) 00. (e) l 0 (f) l (g) 00 0 (h) l.609. Evlute the followig rithms usig your clcultor. Aswer to deciml plces. () 0 ().9 (c) 0. (d) 8 (e) l.0 (f) l000 (g) l 0.0 (h) l 9 (i) (j) 0.. Workig i reverse, fid the vlue of. () ().8779 (c) 0.90 (d) ( ).690 (e) l.7 (f) l 0.7 (g) l 0. (h) l (i).7 (j) 0. Pge 9

14 Numercy Itroductio Before strtig this topic you should review the Defiitio of Log topic. Properties of Logrithms These re the sics of Logs, sed o the defiitio of Logs l 0, 0 l e, 0 l e, 0 l e, 0 From epoets, we kow tht y umer rised to the power of 0 is 0 equl to, ie:, usig the defiitio of Log, 0 From epoets, we kow tht y umer rised to the power is itself, ie:, usig the defiitio of Log, From epoets, Strtig with, usig the defiitio of Log,, usig the defiitio of Log, i epoetil form: The Three Logrithm Lws The Log Lws help simply Log epressios. The first lw: (Product Rule) Proof: A + B AB Note: ll three ses re the sme. Let A the A Let B y the B y + y the AB (Ide Lw ) y I epoet form: AB + y to Log form: AB + y (usig the defiitio) AB A + B Emples: Simplify + Cetre for Techig d Lerig Acdemic Prctice Acdemic Skills Digitl Resources Pge ctl@scu.edu.u [lst edited o 7 Septemer 07]

15 + ( ) 0 Simplify (with o se metioed, the iferece is tht the se is 0) ( ) 8 0 Simplify + y + y ( y) 6y Simplify l + l y+ l0 (Rememer: the turl rithm is se e) l + l y+ l0 The first lw used i reverse is: l( y 0) l(0 y) Give 0.00, 0.77 d , evlute 0 The secod lw: (Quotiet Rule) 0 ( ) ( ) A A B B Note: ll three ses re the sme. Proof: Let A the A Let B y the B A y the (Ide Lw ) y B y A I epoet form: y B A to Log form: y (usig the defiitio) B A A B B Pge

16 Emples: Simplify 0 Simply ( + ) Or i reverse: Give 0.00,, evlute (0 00 ecuse 00 0 ) The third lw: (Power Rule) Proof: let the m ( ) rise oth sides to the power of usig the power of power epoet rule m (or m) usig the defiitio of m m m Emples: Evlute (or simplify) or 0 Pge

17 (0) 0 The Chge of Bse Lw N N This Lw is useful for chge rithm i y se to rithm with se such s 0 or e so tht clcultor my e used. Proof: Let m Usig the defiitio of Log: N m m N tke Log (se ) of oth sides N m usig third lw N m rerrgig N N N N m Emple Evlute 7 Evlute l 7 7 or ltertively l Usig clcultor gives.77 (to d.p.) ( ).09 to d.p. Video Usig Logrithm Lws Pge

18 Mied emples: Evlute or Evlute 6 ( ) Evlute Pge

19 Evlute Write s sigle rithm + y z + y z + y z y z Write i terms of idividul s: l y e y l e l l l + y e l + l y l e e l + l y e Simplify + + Epress s Log with se Pge 6

20 Activity. Evlute the followig without clcultor () () (c) (d) (e) (f) (g) (h). Evlute the followig: () +.8 () (c) + 9 (d) (e) 0 (f) 6 (g) 8 6 (h) 0 0 (i) 7 6 (j) (k) 8 (l) 6 6 (m) () Write s sigle Logrithm: () + c () + y + 7 z (c) y z (d) + c + (f) (e) ( + ) ( 9) (g) + (h) l( ) + Pge 7

21 . Write i terms of idividul s () c (c) ( ) (e) () y y (d) y. Write i terms of idividul s z 8 (f) () () (c) (d) + y+ Pge 8

22 Numercy Solvig Logrithms d Epoetil Equtios Logrithmic Equtios There re two mjor ides required whe solvig Logrithmic Equtios. The first is the Defiitio of Logrithm. You my recll from erlier topic: If umer (N) is writte s power with se () d epoet (e), such s e N the N e. The se,, must e positive umer ( > 0) d ot equl to ( ). The umer, N, must lso e positive (N > 0) For the equtio elow, chgig the Logrithm form to Epoetil form will solve for the vrile. (These were covered i the Defiitio of Log topic) Note: Whe solvig y Logrithm equtios, you should perform check to see tht the swer(s) otied were cosistet with the defiitio of Log. Similr to this, the questio elow is solved usig the defiitio of Log d kowledge of equtios cotiig epoets. ± Cetre for Techig d Lerig Acdemic Prctice Acdemic Skills Digitl Resources Pge ctl@scu.edu.u [lst edited o 7 Septemer 07]

23 Check: As the se of Log must e ( > 0) d ( ), the solutio - is ot cosistet with the defiitio. The solutio to this questio is This type of questio c ecome more comple depedig o the umers. I this emple, clcultor is required. You my eed to review Epoets Equtios with Epoets to d.p. A third type is where the vrile ecomes the epoet. For emple: c e rewritte usig the defiitio of to e which c e solved s epoetil equtio (see elow). However, much esier method of solutio is to use the Chge of Bse Rule. l ltertively l.6 to d.p. The secod mjor ide is sed o equivlece. Put simply: If... M N the M N Note: There must e sigle Log term o ech side of the equls sig. Emple: Usig Log Rule (Product Rule) 0 Emple: l l 6 l l 6 Usig Rule (Power Rule) 6 ± 6 Pge

24 Check: The rithm c oly e foud of positive vlues oly, so the solutio is 6. Some equtios will coti mi of Log terms d umers, there re two strtegies to solve this; (i) Rerrge to get Log terms o oe side d umer(s) o the other, or (ii) Replce the umer with equivlet Log. For emple - is the sme s,, 00 etc. Emple: Solve for : ( + ) Usig strtegy (i) ( + ) ( + ) or Usig strtegy (ii) ( + ) ( + ) ( + ) Mied emples: (i) Solve for (ii) Epress y i terms of : y + 8 y + 8 (iii) Epress y i terms of : l y + l y + 8 y 8 y 8 Pge

25 y l y + l l l y l y e y e y e (iv) Solve for : 6 + 6( + ) + ( + ) 6 6 6( ) ( )( + 9) 0 or 9 Check: Becuse it is ot possile to hve the Log of egtive umer, the solutio is. Video Solvig Logrithmic Equtios Pge

26 Activity. Solve the followig Logrithmic Equtios. () 0 () (c) 0. (d) 7 (e) ( ) (g) (i) ( ) + (f) 9 0. (h) + (j) (k) 9 0. (l) 6 (m) ( ) () (o) 0 (p) 6 (q) 0 (r) l( + ) e 7. Solve for i the Logrithmic Equtios. () () l l 6 (c) ( + ) + 6 (d) (e) + (f) ( + ) (g) + 6 (h) + ( ) (i) 8( + ) 8 (j) ( + ) (k) + (l) ( ) (m) ( ) () l( ) + l l (o) ( + ) (p) +. Solve for y i terms of the other vriles preset. () y + () y + (c) y (d) y+ + + y + (f) (e) (g) l l l y e y+ (h) 7 y (i) l y + l (j) ( y ) (k) y e y + (l) 0 Pge

27 Epoetil Equtios A equtio where the epoet (ide) is vrile or cotis vrile is clled epoetil equtio. A emple is 0. Let s cosider the emple elow: If ivestor deposits $0 000 i ccout tht compouds t % p.., how is it efore this mout hs icresed to $0 000? (Compouded ully) Usig the compoud iterest formul gives: A P( + i) (.0) The first step is to rerrge the equtio to get the form or The tke the of oth sides. Becuse clcultios will eed to e performed, ordiry rithms or turl rithms should e used tkig the of oth sides..0 usig the third lw. rerrgig l. l.0 tkig the of oth sides l. l.0 usig the third lw l. rerrgig l.0 8. This mes tht it will tke 8.yers (9 yers i prctice) for the growth i vlue to occur. This c e checked y clcultig: This emple cotis simple power I this emple, the must e rerrged to oti the power s the suject efore tkig the rithm of oth sides. Pge 6

28 I this emple, there is power s the suject; however, the epoet is more comple th previous emples ( ) Emple: Jo wts to retire whe her superutio fud reches $ She ivests $00 moth (fter t) ito her superutio fud. She ssumes tht the superutio fud will retur 6%p or 0.%pm. How efore her gol is reched? This is Future Vlue of Auity, with r 0.00, R 00 d S ( r) + S R r ( ) ( ) mult oth sides y divide oth sides y dd to oth sides To solve this epoetil equtio, s re required. Pge 7

29 tke Log of oth sides use Log Lw (rouded up) It will e 97 moths or 6 yers moths efore Jo hs eough moey. Video Solvig Epoetil Equtios Activity. Solve the followig epoetil equtios. () 0 ().6. (c) (d).6 00 (e) 6 0 (f). 0.6 (g) 0 (h) (i) 6 6 (j) +. If ivestor deposits $0 000 i ccout tht compouds t 7% p.., how is it efore this mout hs icresed to $00 000? (Compouded semi-ully). Use the compoud iterest formul: A P( + i) Pge 8

30 Numercy Aswers to ctivity questios Topic: Defiitio of Log. Write the followig i rithmic form. () (c) (e) (g) 0. (i) 7.89 e ( mes l) e l 7.89 e () 0 0 (d) 6. (f) (h) (j) e l Write the followig i epoetil form. () (c) (e) (g) 00 () (d).6 0 e l 0 (f) l (h) e l.609 e Evlute the followig rithms usig your clcultor. Aswer to deciml plces..609 () 0 g0.979 ().9 g Cetre for Techig d Lerig Acdemic Prctice Acdemic Skills Digitl Resources Pge ctl@scu.edu.u [lst edited o 6 Novemer 07]

31 (c) 0. g0. - (d) 8 g(p8) (e) l.0 h.0. (f) l000 h (g) l 0.0 h (h) l 9 h(p9) -.97 (i) (j) gpg0. -. Workig i reverse, fid the vlue of. () (c) ().8779 (d) ( ) dd to oth sides 0 divide oth sides y (e) l.7 (f) l 0.7 e.7.8 e (g) l 0. e (h) l l 0.7 e (i) Topic: The Logrithm Lws (j) Evlute the followig without clcultor Pge

32 () (c) (e) 6 6 () (d) (f) ( ) ( ) (g) (h) ( ). Evlute the followig: () (c) ( ) 9 () (d) (0 0 0.) ( 8 ) ( 6 ) Pge

33 (e) 0 (f) (g) (i) (k) (h) (j) (l) Pge

34 (m) () Write s sigle Logrithm: () + c c (c) y z (e) ( y z) yz c + c (g) +. Write i terms of idividul s () c c + + c + + c () + y + 7 z y z y z (d) + + ( + ) + (f) ( + ) ( 9) + ( ) ( 9) ( + ) ( + ) ( ) ( + ) ( + ) ( ) (h) l( ) + l( ) + l e l( ) + l e l e ( ) () y z + y z + y z + y z Pge

35 (c) ( y ) + + y y (d) y ( y ) ( y ) + ( ) + ( y) ( ) ( y ) (e) (f) 8 8 ( ) ( + ) Pge 6

36 . Write i terms of idividul s () ( ) (c) () (d) 7 ( y) + y+ + y + Topic: Solvig Logrithmic d Epoetil Equtios. Solve the followig Logrithmic Equtios. () (c) or (e) ( ) (g) 9 ± 9 ± (i) ( ) ( + )( ) 0 or Checkig; whe -, 8 () (d) 7 (f) (h) (j) to d.p. Pge 7

37 (k) (m) (o) (q) whe, ( ) ( ) 0 + There re two swers of 0 0 or Defiitio (l) () (p) 6 6 ± 6 8 Bses cot e egtive.7 Bses cot e egtive 6 6 ± (r) l( + ) e chge of se Solve for i the Logrithmic Equtios. () (- is ot possile) (c) ( + ) + 6 ( + ) (e) ( 6) 0 6 (0 is ot possile) () (d) l l 6 ( ) l l (- is ot possile).87 to d.p. (f) + ( ) ( + ) Pge 8

38 + 6 (g) (i) ( + ) 8 8 ( + ) + ( + ) (k) + (m) (o) ( ) ( + ) ( + ) (h) + ( ) + ( ) ( ) or 8 8 (j) ( + ) + ( + ) ( + ) ( )( + 8) 0 (-8 is ot possile) (l) ( ) () (p) l( ) + l l l ( ) l ( + )( ) (- is ot possile) (- is ot possile) Pge 9

39 . Solve for y i terms of the other vriles preset. () y + y y (c) y y y (e) (g) (i) + y + + y y y y y y+ y+ y y 8 y l y + l l y + l l l y y l y e y e y e () (d) y + y [ ] y ( ) y y+ + y y y 0 y y (f) l y l e l l y l e l e l y l e y y (h) (j) e y y + + y 7 y 7 y ( y ) y ( ) y y y y 8 Pge 0

40 (k) l y y e yl y l (l) y + 0 y 0 y 0 + y ( + ) + Pge

41 . Solve the followig epoetil equtios. () 0 0 Tke of oth sides 0 Usig the third lw 0.6 or l 0 l 0 Tke turl of oth sides l l 0 Usig the third lw l 0 l.6 () (c) (d) (e) ( + ) l.6 l. l.6 l. l. l l l 0. l l 0. l 0. l l.6 0. l00 0. l.6 l00 l00 0. l l 6 l 0 ( + ) l 6 l 0 l 0 + l Pge

42 (f) ( ) (g) + 0 rerrge to get s the suject (h) 6 0. rerrge to mke 6 the suject l. l 0.6 ( ) l. l 0.6 l 0.6 l rerrge to get s the suject 0 8 l l 8 l l 8 l 8 l rerrge to mke 6 the suject l 6 l 7. l 6 l 7. l 7. l 6. (i) s the ses re the sme the epoets c e equted + 7 Pge

43 (j) + + (+ ) (+ ) 0.07(+ ) l l + l (+ ) l l (+ ) l 0.07(+ ) If ivestor deposits $0 000 i ccout tht compouds t 7% p.., how is it efore this mout hs icresed to $00 000? (Compouded semi ully hlf yerly). Use the compoud iterest formul: A P( + i) 7% p...% per hlf yer A P( + i) ( + 0.0) It will tke.66 hlf yers for $0 000 to grow to $ This is 6.8 yers or, i relity, 7 yers to the ed of the period. Pge