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[.]
Trang 1VAN 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 khi co su khde biet khd Ion Dien hinh Id tri thuc
ve logarit trong 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 trong 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 trong 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
khi thuc hipn phep chia va khai can Trong khi dd, su phat triln cua khoa hpc thdi biy gid ddi hdi cin phai tinh nhSn, chia, khai cSn hieu qua hon Chinh dieu do da thdi thiic Napier sang tao ra PP tinh nhan, chia, cSn bac hai, can bac ba dua tren logarit Tuy nhien djnh nghia khai niem logarit do Napier dua ra hoan toan khac so vdi chung ta biet ngay nay
HVCH, Tn/dng Ogi hpc Su- pham TPHCM
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Trang 2^^ * K
Hinh 1 Hai duang Ihdng song song, dogn SQ, dogn SQ cho tru&c vd cdc diim
do hai diem B, b vach ra
Theo [10], Edward Wright chi ra ring: Napier da tuong tupng hai diSm B va b chuyin dpng tren hai duong thing song song (Hinh 1), trong khi 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 0 nhimg
khoang thai gian bing nhau dilm B vach ra cac diem C, D, E, tuang img vol thoi
, RQ cz dz ez
diem 1,2,3 trong khi do diSm b vg ra 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 ra trSn hai dudng thang theo nhirng
khoang thdi gian bing nhau Napier da chpn dq dai az = 10.000.000 va tao ra nhung
btog tinh logarit cto thiSt cho cac tinh toan cua minh
Nhu v|y, khai niem logarit do Napier xay dung dudng nhu khac biet so vdi khai niem logarit chiing ta biet ngay nay^, do 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 thanh 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 trong dinh nghTa ban diu nay ThSm nira, khdng cd mpt dinh nghia tudng minh nao cho co so ciia logarit Vay, logarit do 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 do minh tao ra 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 2 log„„, b = log„ a + log,
Trang 3• Neu a,b,c,d Id-bon so hgng lien tiep cua mot CSN
ihl 31og„„^í = 21og„,^a + log„„^J vd 3 log„„^ c = 2 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 trong CSN gdm 4 sd hang khi 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 khi tinh Vl0.000.000x5.000.000, Napier dua tren tinh ch4t da chumg minh, dng ldy log„„^ 10.000.000 + log„„^ 5.000.000 = 0 + 6931470 = 6931470 va 6931470-^2 - 3465735 Napier tra bang logarit va tim dupe ket qua 7071068, tuong đi gan vdi ket qua diing
Vdi bai toan thii hai, de tien theo đi chiing tdi ki hieu CSN vdi 4 sd hang sau
a\ b\ c; d trong đ a=14142135, t/=5000000 Rd rang b^ =ậd ; c^ =d-.a, do đ ta ed thi tinh dupe b;c theo cdng thiic b = ^ậd ; 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', do đ cd CSN 14142135, 10000000,
7071068, 5000000
Nhu vay, logarit do Napier tao ra 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 bạ 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 iẹ Vai trd ciia logarit thuc sy da "tiln xa" hon vai trd ciia no trong lich sir Khdng nhthig dupe irng dung rpng rai trong Toto hpc ma logarit con xuit hien trong 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 dọ
2 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 ra đi logarit đng vai trd la mot cdng cu don gian hda cac phep tinh nhto, chia va khai cto thanh cac phep tinh dan gito hem Dudi tac đng 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 ( 0 < O ! ' l ) tac đng vao biSu thirc
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Trang 4x^ y^ 4z (;c, J, z > 0 ) ta biSn ddi dupe nd thanh' 3 log„ x + 2 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 khi ta cin tinh cac gidi ban vd dinh r,0°,oo°; dd la cac gidi han cd dang lim/(;c)"'' trong 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°,<io° nhung phd biln nhit va chung cho ca ba dang tren la ki thuat sir dung logarit Sau day la hai vi du va ldi giai tuong irng:
Vidg I: Tim lim_(sinx)'"'(Gidi ban vd dinh dang 0°) (Tham khao [13])
Ldi giai: Dat y=(sinir)"' Liy logarit nSpe hai vl cd
ln{sinx)
In J = tanx.ln{sinjc)
=-cotx Matkhac |imM!!!li) '"=""'|ira_silli.,|i„,(_eo5x.sinx) = 0-Dodd limlny = 0
Sin X
Vay, lim(sinx)'"' = lim v = lim e'"' = e° = 1
^-•0* i-»0* , - , 0 *
f7rfii2;Timgidihan: lim(sinx)'""" (Gidi ban vd djnh dang 1")
L&igidi: 5) Dat ^=(sinx)"' =[l+(sinx-l)]'"'
to.^ = tanxln(l+(sinx-l)) = ' " < ' ^ < ^ ' " ^ " ' ) ) = M l : ^ ( ! ! ! l ^ J J ! l £ z l
cotx smx-1 cotx , sinx-1 sinx-1 „ sinx-1
Va = smx Dodd lim =0
Trang 5Cudi ciing; lim In yi = 0, nghia 1^; lim A = lim (sin x)'*"^ = e° = 1 ([8], tr.46)
TLT hai vi du tren ta thSy: logarit tham gia vao kT thuat tim gidi han vd dinh dang
r , 0 ° dgdc dp tac dpng vao bieu thiic/(x)^ vabiennd thanh g(x).ln[/(;i:)] Tir dd
thay vi tinh gidi han true tilp, logarit chuyen ham sd y = f{x) ve dang don gian
hon g(ji^).ln[/(x)l va ap dung cdng thiic L'Hospital hoac lim—^^ - = \ de tinh
Mue dich tinh toan dupe thuc hien thdng qua su bien ddi ciia logarit Du chiing tdi khdng dua ra vi du va ldi giai cu the eho dang vd dinh co" nhung ban dpc cd the ap dung kl thuat tuong tu de tim gidi han Ban doc cd the thii tim lim —
^^'i\x)
2.1.2 Tinh dgo hdm cdc hdmsoco dgng y=f[xf^ vd y=f^'[x).f°'{x) f^''{x)
Trong giai tich, ta thudng gap cac bai toan tinh dao ham cua eac ham sd cd dang
y = f{xy'^\ chang ban nhu bai tap sau:
I Bdi 3 Tinh dgo hdm cua cdc hdm so: y = x'
Ldi giai dupe trinh bay bdi tac gia Nguyen Dinh Tri:
i
De y ring ham sd y = x^ khdng thudc dang a'' (vi x khdng phai la hang sd), ciing khdng thudc dang x" (vi — khdng phai hSng sd), do do, mudn tinh y' nh4t thiit phai
X
lay logarit eiia hai ve vakhi dd ed ]ny = ~.\nx
x
Va ^ = - ^ b i ; c + - - - ^ ( l - l n x )
y x^ x X x^
Dodd: y' = ^{l-\nx)=^{\'lnx) ([8],tr.61)
Rd rang, ham sd y = fixy^'^ cho trong bai tap tren cd dac dilm>' = g(x) va y~f{x) khdng la ham hSng nen ham sd j = /(x)^'^' khdng phai la ham sd mii va
cung khdng la ham luy thira, keo theo ta khdng thi tinh dao ham bang cdng thiic thdng thudng dupe Viec l4y dao ham dupe thue hien bang each ISy logarit nepe hai ve eiia
phuong trinh y = fixY^'\ biln ddi vl dang ]ny = g{x).]nf(x) va l4y dao ham theo biln X hai vl Dii khdng true tiep dua ra dao ham ciia ham sd >' = f{xy^'^ nhung logarit cho phep chuyin ham sd ve dang don gian \ny = g[x).]nf{x) va tao dieu kien thuan
lpi cho viec tinh dao ham
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Trang 6Tap chi KHOA HOC DHSP TPHCM
Khdng chi riSng nhirng htoi sd cd dang y = f(xY^'\ ham sd cho bdi cdng X\mcy = f°'(x).f° {x) f°-(x) (a, eIR,Vi = i ^ ) cung cd thi dupe tinh dao ham
thdng qua su tac ddng cua Ipgarit Dudi day la vi du minh chiing dieu dd:
Bdi 3 Tinh dao ham cua ham so y
Ldi giai dua tren hudng din eua tai lieu [8]
l +
x-, x # ± l ([8]x-,tr.54)
Ta cd: Vx * ±1 thi I;-! = Suyraln|>'| = -.[ln|l + x ' | - h i | l - (2)
Liy dao ham hai vS ciia (2) cd: - If 3x'
l - x - V l - x ' "
Ham sd cho ti-ong bai toto trSn khdng cd d?ng y = f°'(x).f°'(x) f°-(x) (a, e R,Vl = i ^ ) nhung ta ed thi chuyin dupe vi y = f'' {x).f°' {x) f°- (x) nhd cac
tinh chit cila luy thua mil sd thuc Logarit da tham gia nhu thi nao tiong kT thuat tinh
dao ham cua ham sd ;; = f°' (x).f°' (x) /_"• (x) ?
Tit vi du tien ta thiy: viec liy logarit tac dpng vao hai vl cua
y = f°' (x)./"' (x) /"- (x) khdng phai dupe thue hien mdt each tiiy y, ma biln doi dupe tiln hanh trSn nhOng gia tri x ma ham sd cd dao ham va f (x)>0
h o j c k (x)|>0, V! = l;n 6 vi dy trSn, tai nhttag gia trj x * ± l ham sd ludn cd dao
ham va > 0 nen viee lay logarit nSpe hai vS ciia |>'| =
1-x^ hoan toan ed the thuc hien dupe Theo dd, mdt each tdng quat logarit tac ddng vao hai vS cua
H=i/, wf'-iAc^f-i/.wr
biln ddi nd thanh hi|;'| = a|ln|y^ (x)| + aj.ln|/j (x)| + + a„ln|/, (x)|, va viSc tinh dao ham cua ham sd ban dau da dupe chuyen vS tinh tdng dao ham ciia cac ham sd don gito ban Nhu vay, logarit tham gia vac ki thuat tinh dao ham ciia cac ham sd dang
y = f°' {x).f"' (x) /"- (x) nhu la cong cy biSn ddi ham sd cho dudi dang tich vS ham
sd don gito hem Thdng qua biSn ddi dd cho phep ta thuc hien dupe myc dich tinh toan
2.1.3 Chuyen cdc hdm mii, liiy thira ve cdc hdm luyen tinh vd bdn luyen tinh
Theo [5], kinh te lupng la mdn toto kinh tS vS do ludng, djnh luong cac mdi quan
he kinh tl va du bao kha ntog phat triln Viec thiit l?p cac md hinh toan hpc, hay don
Trang 7gian la xay dung cae ham toan hpe dien ta cac mdi quan he kinh te anh hudng rat ldn den udc lupng va du bao Trong thuc te de dien ta tde dp tang trudng dan sd, lupng eung tien, viec lam, cac nha kinh te hpc d§ sii dung md hinh hdi quy mu va liiy thira, chang han nhu ham san xuat Cobb-Douglas va edng thiie lai gop"^
Cd nhieu phuong phap ude lupng nhung thudng diing nhat la binh phuong nhd nhat - cdn gpi la phuong phap OLS (Ordinary Least Square) do nha toan hpc Diic, Carl
Friedrich Gauss d% xuat Md hinh hdi quy mu, luy thira eiing dupe ude lupng theo
phuong phap nay Vay cac nha toan hpc, cac nha kinh te hpc da sir dung cdng cu gi va
su dung nhu the nao de bien ddi cac md hinh hdi quy mii, luy thira ve md hinh hdi quy tuyen tinh?
Chung tdi tim thay cau tra ldi trong phan trinh bay d trang 64 va trang 99, tai lieu [5] nhu sau:
Trong II thuyet tien te, tai chinh va ngan hang, ehung ta da biet cong thirc tinh
Iai suat gpp: Y, = Y^[l + r)' (3.20), vdi r la toe dp tang trudng (theo thdi gian) ciia Y
Lay logarit tu nhien ciia (3.20) ta dupe: In}^ =ln7o+/.ln(l + r) (3.21)
Nlu dat /3^=]nYQ ; ^=\n(\+r) ta cd thi vilt (3.21) dudi dang: ]nY,=j3^+/}^J (3.22)
Nlu them ylu td nghi nhien vao (3.22), ta cd: \nY,==/3^+0^.t+U, (3.23) ([5],
tr.64)
• Tir phuong trinh (4.42)^, rd rang quan he giiia Y vdi X; va X3 khdng phai la tuyen tinh Tuy nhien, neu lay Ldgarit hai ve ta dupe:
InY, = \n^,+ /i^\nX^, + /3y[n-X^, +U, = /?„+/?, In^T,,+^3.1n^,,+f/, (4.43) Trong dd ^^ = In y?,, (4.43) la md hinh hdi quy tuyen tinh logarit ([5], tr.99) Tir trinh bay tren ta thay: bien ddi cac ham hdi quy mii Y,=Yo{\+r)', lijy thira X=/i^JC}JC^£''' vl cac dang \nY,=j3^^J3^J va ln}f = /|+j^lnX3,+41n.t;+f^ dupe thuc hien
nhd tac ddng ciia logarit nepe vao hai ve ciia phuang trinh Dii cae md hinh hdi quy
(3.23) va (4.43) chua phai la md hinh hdi quy tuyen tinh, nhung vdi phep dat Y' =\nY,
a (3.23) va dat Y' = \nY,;Xl =\nX^, •,Xl = \nXj_ a (4.43) cho phep ta chuyin chung
ve dang tuyin tinh Viec chuyen nhu vay tao dieu kien cho viec ude lupng cac md hinh tren cd the dupe thuc hien theo phuong phap OLS
Tir dd cho thAy, logarit la mdt cdng cu tdt de chuyen cac ham hdi quy phiic t^p cho dudi dang mil, luy thira ve cac ham don gian hon, ham tuyen tinh hay ban tuyen tinh Thdng qua bien ddi dd viec udc lupng de thue hien hon thay vi udc lupng true tilp tren cae ham hdi quy mii, luy thira
2.1.4 Gidi phuang trinh mil dgng a^^'^ = b i^^'^'=M''* (0<fl^l,6>0)
Tir th\re tl cupc sdng, cd nhieu sir kien dan den viee giai phuang trinh (PT) dang
Trang 8Tap chi KHOA HOC OHSP TPHCM
a'^''' = b (0<a*l,6>0), ching ban:
Tinh sd nam gdi tiln N tir cdng thirc lai kep C = A.{l + r)'' biit sd tiln gdi ban diu
A, lai suit r% mdi nam, tdng sd tiln bao gdm ca lai lln vdn sau N nam
Tinh thdi gian phan ra t cua cac chit phdng xa tir cdng thirc m = m„c"^ biit m„,m la khdi lupng ban diu, khdi lupng cdn lai sau thdi gian t cua chat phdng xa
Ta cd thi giai PT mu a"'' =b, a " " = 6**'' bing kl thuat dua vl cimg ca sd bdi biln ddi "b thanh a'""'"' vk dan din kit qua /(x)=log„6 hojc / ( x ) = g(x)log„6
Tuy nhien, ta cd thi giai PT bing ki thuat sii dung logarit Dudi day la hai vi dy diln hinh va ldi giai tuong img cho hai dang PT mu da neu:
Vidu 8: GUI PT e " ' = 1 0
Ldi giai dupe de nghi bdi tac gia
James Stewart:
Lay logarit hai vS cua phuang trinh
va sir dung (9)^:
ta(e"') = lnl0
5-3x = lnlO
x = i ( 5 - l n l O )
([12],tr.66)
Bdi 2.98: Giai PT 5''=7''([3],
ti.86) Ldi giai: Lay logarit co sd 5 hai ve rdi chia ca hai ve cho 5', ta dupe:
( ^ 1 =log5 7 « x = log,log5 7
([3],tr.ll8)
TCr hai vi du ta tdng quat dupe: logarit ca sd a tac ddng cua hai vl PT a"'' = 6, o""=6'''''' va biSn ddi nd thanh / ( x ) = log„6 va / ( x ) = g(x)log„6 Nhd su tac ddng
dd, thay vi giai tiuc tilp cac PT a"-'^ = b, a'^"*=6"''' ta giai cac PT / ( x ) = l o g , i ,
./^ (^) = g (^) log,, * bSng cae kT thuat giai PT dai sd thdng thudng Dii chua dua ra nghidm
cy the cho PT ban diu, nhung logarit dupe xem nhu mdt phin khdng thi thilu ti-ong ki
thuat giai PT a " ' ' = 6, a " " =/>«''' Qua kT tiiuat giai PT a"''^ = b, a " " =6''"', logarit
ndi b^t vdi ung dung giai hai loai PT dd nhung quan trpng hon la vai trd cdng cu eho phep chuyin viec tim nghiem eua PT cd dang mu vl tun nghiem ciia PT don gito hon
Nhgn xet:
Trong Toto hpc, logarit dupe irng dung dS tinh gidi ban cac dang vd djnh r,0°,co°; chuyen cac ham mu, luy thira vl cae ham tuyin tinh, bto tuyin tinh; tinh dao
ham cac ham sd cd dang ^ = / ( x f " , y=f^[x).f'^'{x) f^-{x] va giai PT mu dang
a' ' = 6, o''"' = 6"''' Thdng qua nhihig ung dyng dd, logarit tiiyc sy ndi bat vdi vai tro edng cu cho phep chuyen viec nghien cim cac biSu thirc phiic t?p cd dang tich, thuong,
Trang 9liiy thira v6 eac bieu thiic don gian han nhd mdi quan he giiia phep nhan va phep edng Thdng qua sir tac dpng ciia logarit, cac bieu thiie phiic tap dupe chuyen ve dang don gian hem va muc dich tinh toan dupe thuc hien tren nhiing bieu thiic don gian do
2.2 Logarit - cong c^t tinh s6 cdc elm so cua mpt so nguyen duong
Trong tinh toan, ddi khi ta can phai xac dinh mdt sd nguyen duong ed bao nhieu chir sd Cd nhieu each tinh sd eac chii sd, cd the tinh bang each dem timg chir sd mdt Chang hgn vdi sd 1357902468, bang each dem ta xac djnh dupe nd cd 10 chit sd Tuy nhien, ta khdng the xac dinh bang each dem cd bao nhieu chii sd cua 2'°''' khi viet trong he thap phan Nhung logarit cho phep lam dupe dilu dd
Gia sir x la mdt sd nguyen duong cho trudc can xac dinh sd eae ehii sd Theo tinh chat ciia sd tu nhien, ta tim dupe mpt sd tu nhien n sao eho 10" < x < 10"*^' (!) Lay
logarit eo sd 10 hai ve cua (1) ta dupe w < log x < w +1 Dieu nay chiing td K = [log x]
Do dd, sd chir sd eiia sd nguyen duang x la [logx] + l
Tir l$p luan tren ta de dang tinh dupe sd chir sd ciia sd 2""" la [log2^'"^] + l = 606 Tuong tu, sd nguyen td Mersenne' A^,.,9g269 = 2'^^*'^'-! cd log(2'^''*^^' - i j +1 = 420921 chii sd Nhu vay, logarit dupe xem nhu la mdt edng eu tdt de tinh sd cac chii sd ciia mpt so nguyen duong bat ki
2.3 Til? logarit
Trong tinh toan, nhieu khi ta can phai chuyen pham vl ciia mot dgi lupng de tien
so sanh, ddi ehieu v^ phan tieh Cd the phdng to kich thudc cua mpt hinh ldn gap m Ian hoac thu nhd n lan dl xem xet Vdi day sd lieu 0,01; 0,1; 10; 100; 1000; 10.000; 100.000; 1.000.000.000 nlu thuc hien giam vdi ti le — thi ta cd day sd nhd hon 10 ldn nhu sau: 0,001; 0,01; 1; 10; 100; 1000; 10.000; 100.000.000 Xet cho ciing ta vin ed mpt day sd phiie tap Neu lay logarit thap phan eae sd tir day sd Ueu ban dau dd ta cd day sd sau: -2; -1; 1, 2, 3, 4, 5, 9 Ro rang day sd ban dau da dupe chuyen ve day sd de theo ddi va de kiem soat hon
Thuc te cho thay, logarit thuc su cd the chuyen cac dai luong cd pham vi rdng hoac qua nhd ve pham vi cd the kiem soat dupe Dieu nay dupe minh hpa bdi thang do
pH, thang dp Richter va thang do decibel - su the hien eu the hda ciia ti le logarit Chiing tdi se phan tich cu the li le logarit qua thang do pH
Theo tai Heu [7]-"7>o«g nuoc nguyen chdt cdng nhu trong bdt ki dung dich ndo luon luon co mat cdc ion IT vd OH " va "nong do ciia cde ion IT vd OH bieu dien du(re tinh axit vd bazo cua dung dieh " Tuy nhien, ndng dp ion H^ cua dung dich
thudng thay ddi trong pham vi rat nhd, khd kiem soat lir 10"'*mo/// cho den
10" moi fl Va theo [7]: ''Moi truong cita dung dich co the bieu dien bdng dgi lucmg thugn l^ hon: dgi luqng chi so hydro pH: pH = -\ogC^ "([7], tr.I19) Vay, chi sd
hydro pH thuan Ipi hon ndng dp ion H"" the hien d chd nao?
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Trang 10Qua phto tich, chiing tdi nhto thiy mdt sd diSm thuto Ipi sau:
+ Thii nhit, theo cdng thirc tinh pH thi pH la gia tri cua ham sd y = - log x, vdi x
dai dien cho ndng dp ion H* va gia tri ciia x thudc doan [lO"";10°] Ma ta biit: ham
sd >' = -logx la ham nghich biln tren khoang (0;+a)) nen moi gia tri x thudc doan
riO"";10°] cd duy nhit mdt gia tri pH tuong irng thudc doan [0;14] va ngupc lai Tir
dd cho thiy pham vi hep [l0"";10°] cua ndng dp ion H* da dupe dua ve pham vi dl theo ddi ban [0;14]
+ Thit hai dua vao chi sd pH ta cung cd the xac dinh dupe tinh axit hay baza cua dung dich Thay vi so sanh ndng dp ion H" 10^'mo/// ta so sanh ehi sd pH vdi 7 (-Iog(l0"') = 7) Theo dd, nlu dung dich cd pH = 7 thi cd mdi trudng hung rinh, dung dich cd pH > 7 thi cd moi trudng baza va dung dich ed pH<7 thi cd mdi tiudng axit + Thii ba, tir chi sd pH hoan toan cd the tinh lai dupe ndng dp ion H* cua dung dich theo cdng thirc C^ =10""'
Nhu vay, khong chi dupe tinh toto trong Toto hoc, logarit edn dupe irng dung de xac dinh chi sd pH cua dung dich Tir img dyng tinh pH dd ta thay logarit ndi bat vdi vai tro cdng cy cho phep chuyen dai lupng cd pham vi nhd ve pham vi cd thS kiSm soat dupe
Trong Itiii ndng dp ion H^ dai dien cho dai lupng cd pham vi nhd, hep thi eudng
dp cac trto ddng dat, cudng dp ciia toi thanh la nhirng trudng hop dien hinh eho dai lupng cd pham vi tucmg ddi rdng Chang ban, dp manh ciia cac trto ddng dat dao dpng
trong khoang IQ eho den 800,000,OOOIo vdi Io la bien dp dao ddng be ban \pm trSn may
do dia chto dupe do bing dia chan kS dat xa each tam chin 1 OOkm
Logarit dupe img dung de xae dinh dp chan dpng eua trto ddng dit va dp to nho ciia am thanh theo cac cdng thirc:
Dp chin dpng ciia cac trto ddng dit: M = log —(don vi dp Richter), trong dd lo la
bien dp dao dpng chuan, I la bien dp dao ddng dupe do bing dia chin kl dat xa each tam chan 100km
• Cudng do toi thanh: I = 10log —(dan vi decibel), trong dd I la nang lupng truyen di bdi sdng am trong mdt don vi thdi gian va qua mpt dan vj dien tich bl mat vudng gdc vdi phucmg truySn (don vi do la W/m^); Io la cudng dp cua am d nguSng nghe (/„ = IO"'=B'/m')
Lpgarit khdng chi don thuan dupe irng dyng de tinh dp pH, dp dp chin ddng ciia cac trto dpng dat, do dp to nhd cya am thanh, ma qua cac ihig dung dd logarit noi bai vdi vai trd cdng cu chuyen cac dai lupng cd pham vi qua hep hoac qua rdng vl pham vi