Tap chl KHOA HOC OHSP TPHCM Nguyin Viit Hiiu VAN DE DAY HOC LOGARIT TRONG CHl/ONG TRINH TOAN PHO THONG VA NHUTVG DIEU CAN BIET VE LOGARIT NGUYEN VI^THI^U* TOM TAT Chuyin hoa suphgm tao dieu kien cho n[.]
Tap chl KHOA HOC OHSP TPHCM Nguyin Viit Hiiu VAN DE DAY HOC LOGARIT TRONG CHl/ONG TRINH TOAN PHO THONG VA NHUTVG DIEU CAN BIET VE LOGARIT NGUYEN VI^THI^U* TOM TAT Chuyin hoa suphgm tao dieu kien cho ngudi hoc tiep can nhanh vd cd he thong cdc tri thuc da duoc nhdn logi thira nhdn Tuy nhien qud trinh dd ldm cho tri thirc khong cdn gidng nhu ngudn gdc ban ddu cua no, doi co su khde biet khd Ion Dien hinh Id tri thuc ve logarit chuang trinh Todn phd thdng hien hdnh Voi mong mudn tim lgi nghia vd vai trd eho ddi tu(mg logarit, bdi viet gioi thiiu sw xudt hiin cua nd lich su vd nhitng vai tro cdng cu qua ede ung dung noi bat Tir khoa: logarit, nghia cua tri thire, Heh sir Toan ABSTRACT The issue of teaching logarithm in high school mathematics syllabus and what to know about logarithm The pedagogical transfer has brought learners opportunities to approach quickly and systematically the knowledge that has been acknowledged by all human, beings However, that process has made the knowledge on longer the same as its origin; in fact, there're sometimes wide disparities A very typical example is the knowledge about logarithm, which has been presented in the current high school mathematics syllabus Aiming to retrieve the meanings as well as the roles of logarithm, the article will discuss the appearance of logarithm in history and its main roles as a tool through outstanding applications Keywords: logarithm, meanings of the knowledge, the history of maths I Vai n^t SO" lirp^c ve Ijch sir xuat hi^n khai ni^m logarit Logarit dupe John Napier' (1550 - 1617) gidi thieu ddu tien tac ph4m "Mirifiei logarithmorum canonis deseriptio" vao nam 1614, sau 20 nam nghien ciiu Di;a tren y tudng "nhan hai sd theo cdng va trir" cua phuong phap (PP) prosthaphaeresis^ cd trudc dd Tuy nhien, PP prosthaphaeresis chiia dung nhieu bat lpi thuc hipn phep chia va khai can Trong dd, su phat triln cua khoa hpc thdi biy gid ddi hdi cin phai tinh nhSn, chia, khai cSn hieu qua hon Chinh dieu da thdi thiic Napier sang tao PP tinh nhan, chia, cSn bac hai, can bac ba dua tren logarit Tuy nhien djnh nghia khai niem logarit Napier dua hoan toan khac so vdi chung ta biet HVCH, Tn/dng Ogi hpc Su- pham TPHCM 55 Tap chi KHOA HOC OHSP TPHCM ^^ Sd 50 nam 2013 *K Hinh Hai duang Ihdng song song, dogn SQ, dogn SQ cho tru&c vd cdc diim hai diem B, b vach Theo [10], Edward Wright chi ring: Napier da tuong tupng hai diSm B va b chuyin dpng tren hai duong thing song song (Hinh 1), dilm B chuySn dpng thep mpt chilu nhit dinh tren ducmg thing dai vo han vdi t6c dp khong doi, bat dau tit A thi dilm b chuyin dpng tir a trSn doan thing az vol tdc dp giam dan nhimg khoang thai gian bing dilm B vach cac diem C, D, E, tuang img vol thoi , RQ cz dz ez diem 1,2,3 diSm b vg cac diem c, d, e, thoa -^7;=—=—=—••• ' ' ' " SQ az cz dz vol doan thing SQ va dilm R tiiupc dp?n SQ cho trudc Napier da djnh nghia: /4C=!og„„p(cz) vdd cz = Sin^i Al>AQ%f^p{dz) vdi dz = Sin^2 AE=\o%i„i,(ez) vdi ez = Sinfla Tuong tu cho cac diSm khac ma B va b vjch trSn hai dudng thang theo nhirng khoang thdi gian bing Napier da chpn dq dai az = 10.000.000 va tao nhung btog tinh logarit cto thiSt cho cac tinh toan cua minh Nhu v|y, khai niem logarit Napier xay dung dudng nhu khac biet so vdi khai niem logarit chiing ta biet nay^, la sir liSn he giiia cac phan tii cua cap s6 cpng (CSC) va cac phin tu cua cip s6 nhan (CSN) Logarit biln ddi cac phin tir ciia CSN phan tii cua CSC tuang ling Tuy nhiSn, khong co mpt djnh nghia logarit mpt so thuc duong bit ki cho trudc, cung nhu khdng cd mpt mdi liSn he gi vdi luy thira mu so thy:e dinh nghTa ban diu ThSm nira, khdng cd mpt dinh nghia tudng minh nao cho co so ciia logarit Vay, logarit Napier xay dung duac sir dung dS lam gi? Tinh chit nao cua khai niem logarit da dupe thiit lap? NghiSn Cliu [10] ehung tdi thiy; Napier da chitng minh mot s6 tinh chit quan trpng cua khai ni$m logarit minh tao Cu thi nhu sau: Neu a,b,c,d Id bon sd eua mdt CSN thda -=— b d th' log„„ a - log,„, b = log,.p e - log„„, d • • Neu a,b,c Id ba sd hgng lien tiep cita mgt CSN 'hi log„„, b = log„ a + log, 56 Tap chi KHOA HOC DHSP TPHCM • Neu a,b,c,d Nguyin Viit Hiiu Id-bon so hgng lien tiep cua mot CSN ihl 31og„„^i' = 21og„,^a + log„„^J vd log„„^ c = log„„^ fi + log„„^ u Theo [10] va 114], Napier da kiem ehiing dupe tinh uu viet ciia logarit thdng qua cae bai toan: tinh trung binh nhan cua hai sd 10.000.000, 5.000.000 va tim sd hang thii hai, thii ba CSN gdm sd hang biet sd h^ng dau 14142135 va sd h ^ g cudi 5.000.000 Napier khang dinh rang: Tinh theo logarit de dang hon each tinh thong thudng Cu the tinh Vl0.000.000x5.000.000, Napier dua tren tinh ch4t da chumg minh, dng ldy log„„^ 10.000.000 + log„„^ 5.000.000 = + 6931470 = 6931470 va 6931470-^2 - 3465735 Napier tra bang logarit va tim dupe ket qua 7071068, tuong ddi gan vdi ket qua diing Vdi bai toan thii hai, de tien theo ddi chiing tdi ki hieu CSN vdi sd hang sau a\ b\ c; d dd a=14142135, t/=5000000 Rd rang b^ =a^.d ; c^ =d-.a, dd ta ed thi tinh dupe b;c theo cdng thiic b = ^a^.d ; c^^d'xt Nhung Napier tinh theo each dua tren phep edng, nhan hai va chia ba, cd su ho trp eiia bang logarit, 21og,',„rf + log„„,fl 2x6931470 + (-3465735J ' , , , , ]og„,^c = —= ^ ^« 3465735 va tra bang logarit ong tinh dupe c = 7071068 Tuong tu 6»10', dd cd CSN 14142135, 10000000, 7071068, 5000000 Nhu vay, logarit Napier tao nham myc dich de don gian hda cac phep tinh nhan, chia, can bac hai, can bac ba theo cac phep tinh don gito hem nhu cdng, trit, chia hai va chia ba Du tinh toto da dupe cai thien nhung ca so logarit chua thuc su tien lai, bang li thuylt toan hien dai ngudi ta chiing minh duoc log,^x=10'.log| -^ Song vdi nhiing uu diSm vupt trdi, logarit da tao hiing thii cho nhieu nha toto hpc nhu Henry Briggs (1561-1630), Nicolaus Mercator (1620-1687), Leonhard Euler (1707-1783), nghien ciiu sau va rdng ban ve logarit Cling vdi sy phat triSn cua khoa hpc, Toto hpc da phat trien rat nhanh va logarit ciing khong phai la ngoai ie Vai trd ciia logarit thuc sy da "tiln xa" hon vai trd ciia no lich sir Khdng nhthig dupe irng dung rpng rai Toto hpc ma logarit xuit hien cac cdng thuc tinh d cac bd mdn khoa hpc khac Chung toi xin diem qua vai ling dyng cua logarit va cac vai trd cong cy dupe thi hien qua nhitag iing dyng Vai tro cfing cu ciia logarit qua mot so ling dung 2.1 Logarit - cdng cu dan gidn hda cdc phep tinh phirc tgp Nhu ta biit, tit Idii ddi logarit ddng vai trd la mot cdng cu don gian hda cac phep tinh nhto, chia va khai cto cac phep tinh dan gito hem Dudi tac ddng ciia logarit cac bilu thiic cho dudi dang tich, thucmg, liiy thira dupe dua vl cac biSu thiic don gian Ro rtog, sir dyng logarit ca so a ( < O ! ' l ) tac ddng vao biSu thirc 57 Tap chi KHOA HOC OHSP TPHCM Sd 50 nam 2013 x^ y^ 4z (;c, J, z > ) ta biSn ddi dupe nd thanh' log„ x + log„ y + — log„ 2, thay vi tinh nhto, liiy thira ta cd thS tinh toto dya tren phep cdng cac logarit Theo tien trinh phat trien cua Toto hpc, vai trd dd cua logarit vin tilp tyc dupe kl thira va cd phat triSn Vai trd cdng cu dan gito hda cac bieu thiic phiic tap cho dudi dang tich, thuang, luy thira ths hien qua cac irng dung sau ciia logarit: a) Tinh cac gidi han vd dinh dang r,0*',oo" b) Tinh dao ham cac ham sd cd dang y=f(xf" va y=f^ i")/^' {^l-ZT'('^) e) ChuySn ham mu, luy thira vS ham tuySn tinh hay ban tuyen tinh d) Giai phuong trinh mil dang a^^'^ = b, a^^^^ =b^^^^ Chung tdi khao sat ey thS timg img dyng tren cua logarit qua cac muc sau: 2.1.1 Tinh cdc gi&i hgn vd dinh dgng l",0",co'' Trong giai tich ddi ta cin tinh cac gidi ban vd dinh r,0°,oo°; dd la cac gidi han cd dang lim/(;c)"'' dd a co ihl hilu han hoac vd ciing Trong trudng hpp 1" thi lim/(x) = l va limg(x) = co, chtog hjn lim In— , tuang tu cho cac trudng hop 0°, =0° NghiSn cim cac tai lieu tham khao, chung tdi thiy cd nhilu kT thuat tim gidi han ciia cac dang vd dinh r,0°, thi cd moi trudng baza va dung dich ed pH