Chapter 2. LOGARITHMS
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1 Chpter. LOGARITHMS Dte: A. INTRODUCTION At the lst hpter, you hve studied bout Idies d Surds. Now you re omig to Logrithms. Logrithm is ivers of idies form. So Logrithms, Idies, d Surds hve strog reltioship. Look t these illustrtios: i the rithmi form: i the rithmi form: i the epoetil form: i the epoetil form: Notes: - i this hpter, we use the sietifi lultor - i some outries: b is writte s b - rithms to bse 0 is usully writte s: b 0 ( ot: b ) - Some of epoetil lws re used i this hpter So we got the first lw: b b Where: bse d, > 0, b > 0 EXERCISE Covert to rithmi form or epoetil form: p 9. 9 y. m u 0. v w Logrithms, SMA PAHOA
2 B. GRAPHIC OF LOGARITHMS Dte: To fid grphi of rithms, we drw it mully or usig the omputer. Grph of Log Grph of "".. "" Grph of 0 Grph of Grph of EXERCISE Without usig lultor, ordig to the grphi bove, estimte the vlue of: (up to deiml ples).....?. 0.?., 0, 0 9.?, Logrithms, SMA PAHOA
3 C. GRAPHIC USING THE COMPUTER Dte: There re my softwres we use to plot or mke grph of rithm (d lso other mth grphs) suh s Mirosoft Eel, Grphl, Grphmti, Mthemti, FXGrph, Skethpd, Mth Mehi, et. Eh softwre hs speifi futios ompre with other. Below is the wy to mke the grph of d by usig Mirosoft Eel.. To fid grph of : Ope Mirosoft Eel - I olum A row (ell A), type: - At ell A to A9 plese type the umbers you eed (ep.,, 0, 0, 0, 00, 0, d 00) - I olum B row (ell B), type : - t ell B, type: (A) the eter - opy-pste ell B to ell B9 - blok olum A d B, mke lig etered Blok olum A d B Clik: Isert Chrt XY (Stter) Choose hrt sub-type t the eter (stter with dt poits oeted by smoothed lies). Clik: Net twie At Leged plese ubloked Show Leged At Titles plese type Grph of Log or lter Grph of Log ( mes ) To fid grph of : - I olum E row (ell E), type: - At ell E to E9 plese type the umbers you eed (ep.,,,,,,, 0) - I olum F row (ell F) type: (mes ) - t ell F, tipe: (E)/() the eter - opy pste ell F to ell F9 - blok olum A d B, mke lig etered - Blok olum E d F - go to step Ple both of grphs properly by plig them below the two tbles, resize them whe eeded. At ell H: type your me At ell H: type lss..... At ell H: type the dte August 009 Clik Prit Preview, mke sure ll grphs re show. Plese prit your work o piee of pper.. Clik: Fiish ASSIGNMENT AT HOME: Mke two grphs of d 0. usig Mirosoft Eel. Plese tke te dt of rdomly d the -rge is betwee d 00. (emple: the -iput re,, _, _, _, _, _, _, _, d 9_). The ssigmet must be olleted before: August 009 Logrithms, SMA PAHOA
4 D. SIMPLE LOGARITHMIC EQUATIONS Dte: We lredy kow, tht if b the b d if b the b We use tht rule to do these simple equtios;. Fid the vlue of if Solutio:. Fid the vlue of if ( ) Solutio: ( ) ( ) ( ) 9 EXERCISE Without usig the lultor, fid the vlue of i the followig: ( ).. ( ). ( ). ( 9). 9. ( ). ( ) 0. 0 ()... ( ) 0. If d y, evlute y Additio: do lso the eerise from Buku Kerj, pge - Logrithms, SMA PAHOA
5 E. COMMON AND NATURAL LOGARITHMS Dte: COMMON LOGARITHMS Logrithms to bse 0 (te) re lled Commo Logrithms. We ofte write 0 s (or just lg i some outries). For emple 0 d 0 (-) re bbrevited s d (-) respetively. Commo rithms be evluted diretly usig sietifi lultor. Rell tht by the defiitio of rithm: b b 0 d sure tht 0. NATURAL LOGARITHMS Besides bse 0, other importt bse is e. Observe tht will pproh erti vlue s beomes very lrge (up to ifiity). This limitig vlue is deoted by e d e, (to deiml ples) Logrithms to bse e re lled Nturl Logrithms. Nottio e is ofte bbrevited s l. So e l d e re writte s l d l respetively. Like Commo Logrithms, Nturl Logrithms lso evluted usig sietifi lultor. By defiitio: b b l b e or, d sure tht l e Emple Solve the followig equtios: ) 0 Solutio: b) e ) 0 0 ( ) 0 0,,, b) e l e l ( ) l e l ( ). l, 9, 9, 9, 9 EXERCISE Solve the equtios: (up to deiml ples). 00. e.. l... e l l. e l. l 9 Logrithms, SMA PAHOA
6 F. LOGARITHM OF A NUMBER Dte: Before we hve lultor, sietist used tble of rithm to fid the et vlue of rithm of umber. Below is the emple of rithm tble: N Wht does it me? How do we use it? The tble shows four sigifit figures of rithm of umber (i olum N). If tht umber is greter th 0 d below 00, the you must dd (emple:, 0, d,,9) but if tht umber lies betwee 00 d 000, just dd (emple:,), d so o. After sietist foud sietifi lultor, the we use it to fid the very et vlue of rithm of y umber. But ll lultors re desiged to bse 0 oly. It fid the vlue of,, 009, d so o beuse the bse of them is 0. 0, But if we wt to fid the vlue of 0,, we must use the rithms lws d 0 hge them first ito,,, 0, et. Emple Fid the et vlue of: (up to deiml ples) ) b) 00 ) 0, d) Aswer: ), b) 00,0 ) 0, -0,9 d), 0 EXERCISE By usig sietifi lultor, fid the et vlue of: (up to deiml ples). 0, , Logrithms, SMA PAHOA
7 G. LAWS OF LOGARITHMS Dte: For, > 0, b > 0, d > 0 :. b b. m b m. b.. (b.) b +. m b. b.. b b d. b b. d Emple usig lws - Evlute the followig: m b b 9. b b b b b b (b.) b +.? Let. 0,? 0, 0, 0, b d b d.? let.? let.? 9 9., 0,?,, 0, 0, do eerise pge o,,, see Mtemtik Iovtif pge -9 see Mtemtik Iovtif pge 9-0 EXERCISE Pge : Pge : do eerise pge o.,,, ) ) ).. ( ) ) ( ) ) ).. ) if 0,0 d 0, lulte? ) 900 Logrithms, SMA PAHOA
8 Emple usig lws - Evlute the followig: m b.? m. b.. Dte: m m b b. b b..?..?.....?... 9? 9...?. do eerise pge o,,, 0 do eerise pge o.,,, Additio problems., EXERCISE Pge : Pge : Additio problems:.. 0, 0, , if the? if 9? 0, 0,. Logrithms, SMA PAHOA
9 Emple usig lws -9 Evlute the followig: b b. if m d evlute d. m m b b b b. if p d q.? evlute d p... q q.? Dte: or 9...? if p evlute p.? do eerise pge o.,,, do eerise pge 9 o.,,, EXERCISE Pge : Pge 9:.. 0, if epress ) b) 9.. 0, Logrithms, SMA PAHOA 9
10 H. LOGARITHMIC EQUATIONS Dte: A equtio tht otis rithm of vrible qutity is lled rithmi equtio. Logrithmi equtio geerlly be solved usig the followig property: For two rithms of the SAME BASE, if M N the M N Emple Solve the followig equtios:. ( ) b.. d. + ( + ) Solutios: Remember if b the b. ( ) b. OR:. + 0 ( ) ( ) (rejeted beuse o b, ot be ) d. 0 0 ( is rejeted, it used : ) beuse o b, b must be > 0 (positive) EXERCISE 9 Solve for i the followig questios: (t home, do lso eerise pge 0- from Buku Kerj Mtemtik). ( ) Logrithms, SMA PAHOA 0
11 I. APPLICATION OF LOGARITHMS Dte: Mybe you hve severl questios like these: - Why do we hve to ler Logrithms? - Wht is is used for? To swer them, see these illustrtios: - if fid? - if. 000 fid? Emple. Fid if If we wt to fid et umber of tht mkes equtio:, we must use the lultor. The vlue of lies betwee d. Why? Beuse < < the we got,..... To fid the et vlue of we eed sietifi lultor d theory of rithms. Here s the first wy: overt ito b ordig to formul: b ( b b ) the, 0 (up to deiml ples), 0 You hek with lultor:? seod wy:? Give ottio o both sides: m. ( b m. b ), 0. Fid the vlue of if. 000 Solutio: give ottio o both sides: , 0 (up to deiml ples) 0 0, 0, 0 [ you hek with lultor:, 0? ] Short wy: , 0 EXERCISE 0 Use the lultor to fid the vlue of : (up to deiml ples) ( + ) , Logrithms, SMA PAHOA
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