Phương Trình, Bất Phương Trình Hữu Tỉ Vô Tỉ Mũ Logarit

512.9 PH561T •4 (CB) • ThS PHAN VIET BAC MHAN - CN LE PHUC LUf J f HAT l»HI]liSS •nrnCo ^ D U N G CHO HS GIOI THI TRirOfNG C H U Y E N ^ O N THI THPT QUOC GIA (2trong1) I^NrtA'XUAT BAN • C QUOC GIA HA NOI 5^1 g TS LE XUAN S d N (CB) • ThS PHAN V I E T BAC ThS TRAN NHAN - C N LE PHUC L Q HUUTI VO Tl MO LOGARIT ^ D U N G CHO HS GIOI, THI TRI/flfNG CHUYEN y ^ O N THI THPT QUOC GIA DTR: N H A X U A T B A N D A I H Q C Q U O C G I A H A NQI NHA XUAT BAN DAI HOC QUOC GIA HA NQI 16 Hang Chuoi - Hai Ba Triing - Ha Npi Dien thoai: Bien t$p - Che bkn: (04) 39714896; Quan ly xuat ban: (04) 39728806; long bien tSp: (04) 39715011 Fax: (04) 39729436 * * * Chiu trdch nhi^m xuat ban: Gidm doc - Tong bien tdp: T S P H A M T H I T R A M NHA SACH HONG A N Trinh bay bia: NHA SACH HONG AN ban: Che tdp: Bi^n VAN ANH - PHUONG ANH ^' Doi tdc lien ket xuat ban: NHA SACH HONG A N SACH L I E N K E T PHaONG TRINH - BAT PHUONG TRINH HLJfU TJ, VO TJ, MU, LOGARIT Ma s5: 1L - 99OH2015 In 2.000 cudn, kh6 17 x 24cm tgi COng ti Cd phin VSn h6a VSn Lang Dia chl: S6' Nguy§n Trung Tri;c - P5 - Q Binh Thanh - TP Ho Chi Minh So xua't bSn: 351 - 2015/CXB/4 - 74/DHQGHN, 09/02/2015 Quyg't djnh xuS't ban s6: 120LK-TN/QD - NXBOHQGHN In xong va nOp liAi chieu quy II n3m 2015 Cac em hoc sinh than men! Phuomg trinh, bat phuang trinh la nhOng noi dung can ban chuong trinh toan ph6 thong Co dugc ky nang t6t viec giai phuang trinh, bSt phuang trinh se khong nhung gop phan quan dS hinh va phat trign nang lire giai quyet van de ciia hoc sinh ma giup cac em dat k6t qua t6t nhtjng ky thi quan nhu: thi vao truang Chuyen, thi dai hoc, thi hoc sinhgioi cac d p Vai muc dich ay, chung toi bien soan cuon sach nham cung cap cho ban doc mot he thdng bai tap phong phu, da dang vai nhi§u bai mai la va cac phuang phap giai hieu qua ve phuang trinh, bdt phuang trinh Noi dung ciia cu6n sach dugc trinh bay ba chuong: Chmmg de cap den phuang trinh bat phuang trinh dang da thuc va huu ty; ChiroTig de cap den phuang trinh, bat phuang trinh v6 ty; ChiroTig va Chirong theo thu tu de cap den phuang trinh, bat phuang trinh mu va logarit Trong tung muc, tung phuang phap deu c6 cac vi du minh hoa tieu bieu; c6 phan bai tap de ban doc ren luyen; c6 phin huong din giai bai tap sau de ban doc tham khao, so sanh vai lai giai cua minh Sau moi chuong deu c6 phan bai tap tong hgp, phan Ian la bai tap hay va kho Chung toi hy vgng rang cuon sach "Phuang trinh, bat phiromg trinh va phuang phdp gidi" se thuc su huu ich cho cac em hoc sinh cung nhu cac thay, CO day Toan a truang thong Du da hit sue c6 ging qua trinh bien soan, nhung bg sach kho tranh khoi nhung thieu sot nhat dinh Cac tac gia chan cam an y kien dong gop cua cac thay giao, c6 giao va cac em hoc sinh gan xa de Ian tai ban bg sach se dugc hoan thien hon Mgi y kien dong gop cho tac gia xin quy ban dgc gai ve: nhasachhongan@hotmail.com CAC TAC GIA ChUctng PHLfdNG TRINH, BAT PHtTdNG TRINH HlTU TI §1 TAM THlTC, PHirONG TRINH, BAT PHlTOfNG TRINH BAG HAI 1) DIU cua tarn thipc bac hai Tom tit ly thuyet 7.7 Dinh ly ve ddu cua tarn thuc Cho tarn thuc bac hai f{x)^ax^+bx + c,a^Q Dat A = b^ - 4ac Khi do: NSu A0 vai moi xeR; NSu A = thi af (x)>0 voi moi x^2a Ngu A>0 thi af{x)>0 vai moi x G ( - O O ; X , ) U ( X ; + C O ) va af(x);-12xv • D o d o \ +{\2y-n)x - -^ =-llx'+(ll-12>;)x-I0/+10>; + \0y^ -\0y + A = De CO gia t r i Ion nhat ciia A t h i phuong trinh c6 nghi?m T a c6 A > < » - / + ; / + 121-44^>0 l l V 495 495 74 22 121 74 ' D o ^ < y- y + — V + < 27 11 11 37 296 148 148 27 25 11 Dku dang thuc xay k h i x = — ; v = — : z = • ^ ^ 74 37 74 495 0V a y gia t r i I o n n h i t ciia A la 148 V I d u Cho hai s6 thuc x,y thay d6i thoa man x^ +y^ = \ T i m gia t r i Ion nhat va nho nhat cua bieu thuc P = • + 2xy + 2y (Di thi dgi hoc khoi B 2008) L&igidL Voi = ta CO = nen P = X 2fx^+6xv) Vai>;^0,datr = - t a c y Dodo P[t'+2t 2(x^+6xy) P^-^ - ^ =- ^ ^ = x'+2xv + / l + 2xy + / + 3) = 2t^+\2tc>{P-2y+2{P-6)t 2/^+12^ • t^+2t + + 3P = , _ ^.l V a i P = 2, phuong trinh c6 nghiem t = — Voi P^2, phuomg trinh c6 nghiem vachi A' = - P ' - P + > o - < P < P = ?> )&h\ = —F=,y = ^ = hoac x = — = , j j ; = - VTo • 3 P = -6 x = — v = — p = hoac x = — i = , y VTI 713 • Vl3 = — = Vn Ket hop lai ta c6 gia tri nho nhdt cua P la - , gia tri Ion nhdt cua P la Vi du Cho a,b^O Tim gia tri nho nhdt cua bi^u thuc b P = a' +b' +^ + - - a a L&igidL Xem P nhu la mot tam thuc bac doi voi bien b Taco P = b' +2b— + -K + +a = b+ 2a la 4a' 4a' + •4a' 4a' D4U bang xay +a + a'>2 -.a' \4a- b= 2a a' = 4a' Vi du Cho cac so duong a, b, c thoa man a + b + c = Chung minh rang a + ab + 2abc< — Laigiau Tugiathiet = - a - c.Tac6 9 a + ab + labc < — • o a + a ( - a - c ) + 2ac(3 - a - c) < — Dat / ( a ) = (2c + l)a^ +(2c^ - c - ) a + ^ > Tachung minh / ( a ) > Ta CO / ( a ) la mot tam thuc bac hai c6 he so ciia a' la 2c +1 > 0, va lai c6 A = ( c ' - c - ) ' - ( c + l ) = ( c - l ) ' ( c ' - c - ) < < c < Tir d o / ( a ) > Dau bang xay a = — ;6 = l:c = — 2 Vi du Cho s6 thuc a,6,c,£/thoa man: + 6^ = 1; c - ; + 1)^ - - 4;; + > O 2>; < >; < Khi y = A thay vao phuong trinh ta c6 x = Vay X = thi dat gia tri Ion nhdt Bai tap phan 1.1 +z =\ loa man Cho x,y,z thoa man < [2x'+3/+4z'=3 Tim gia tri Ion nhdt, gia tri nho nhat cua y Cho so x,y,z thoa man 1'xxy++y_yz+ z+=zx5= Chungminhrang \'z = - x ( > ; + z ) J>' + z = - x j +z =5- x xjr-: ,«f c or - (5 - x ) / + x^ - 5x + = z la hai nghiem cua phuong trinh Phuong trinh c6 nghiem A' > (5 - x)^ - 4(x^ - 5x + 8) > " ' ^ -3x^+10x-7>0 l n > , I n > l n > = > — > — > In3 ln5 r ln6 19-7^-2-6^ 6' > voi moi x > Suyra f'(x)>^^^ hi5 In5 (7^+2)(6^+19) 6'+19 d6ng biSn tren [0;+oo) ma /(1) = nen phuong trinh da c6 r r+2 Suy f{x) nghiem x-l Vay phuong trinh da cho c6 nghiem nhat x-l V i du Giai b i t phuong trinh sau xlog2 JC > ( x - l ) ^ L&i gidL *) DiSu kien x > Do (jc-1)^ > nen ta CO jclog2 X > => log2 JC > JC > Xet ham s6 / ( x ) = xlog^ x - ( x - l ) ^ x > Ta c6 / ' ( x ) = log2X +In- i2- - ( x - l ) , r(x) = X - ^ - < - ^ - < In In Do do, phucmg trinh f{x) = c6 khong qua nghiem Ta cung thSy ring / ( I ) = / ( ) = nen phucmg trinh f{x) = c6 dung nghiem x = 1, x = D e t h i y t r e n c a c m i ^ n [l;2],(2;+co) t h i d d u c u a / ( x ) khongddi Thu true t i l p , ta thdy / ( x ) > 0,x e [1;2] va / ( x ) < voi xe(2;+oo) Vay nghiem cua bat phuong trinh da cho la x e [1; 2] ^ • Bai tap phan (x —1)^ Bai Giai phuong trinh 81og,^^ ^ = x^-18x-31 2x + l Bai Giai phuong trinh log2 (2" + 4) + log3 (4^^' +17) = 274 I- Bai Tim so thuc m dk phuong trinh sau c6 nghiem thuc doan (/«-l)log^(x-2)^+4(/«-5)log Bai Giai phuong trinh 4(x - 2)[log2 (x • + 4m-4 =0 - 3) + log, (x - 2)] = 15(x +1) Bai Giai hit phuong trinh log, | l + 2yJx^-x + 2^ + log, (x^ - x + ? ) < Bai Giai bdt phuong trinh + 3''-2x-l Bai Giai phuong trinh Vx + SyJT^ >Q = log, ((3 - x ) ' (2x +1)) .) t ^ ' , ^ tf a H u a n g dan giai bai tap ph'an Xet phuong trinh log 2x + l = x^ -18x - 31 Dieu kien xac dinh: • x> — x^l Ta c6: - logjCx - ) ' - Iog2 (2x +1)] = (x - ) ' - 8(2x +1) - 24 « (x - ) ' + log^ (x - ) ' = 8(2x +1) + 24 + log^ (2x +1) « (x - ) ' + logj (x -1)^ = 8(2x +1) + log2 8(2x +1) Xet ham so / ( O = / + logj / tren (0; +oo) Ta c6: f'(t) = 1+ /in > voi moi />0 Suy / ( d6ng biSn tren (0; +oo) Phuong trinh ban dku chinh la: f[ix-\y] = f [8(2x + ) ] « ( x - ) ' = 8(2x +1) x - x - = « "x = - V 2 _x = + 2>/22 ' : - - Ta thay rang cac nghiem tim dugc deu thoa man diSu kien ban dku Vay phuong trinh da cho c6 hai nghiem la x = + 2V22,x = - 2V22 [2^+4 = 2" Xet phuong trinh log^ [2" + 4) + log^ (4'^' +17) = Dat , taco: [4^"'+17 = 3' Tu thay vao phucmg trinh da cho ta dugc: 4(2" - 4)^ - 3^'° +17 = Xet ham s6 / ( a ) = ( " - ) ' - ' " " + tren M, taco: / ' ( a ) = ( " - ) - " - l n + 3'-"-ln3>0, V a e M Suy f{a) d6ng bifin tren M Mat khac, / ( ) = nen f{a) = c6 nghiem nhat a = » " = ' - = « x = Vay phuang trinh da cho c6 nghiem nhat x = Phuang trinh da cho tuang duang vai: ( w - l ) l o g ^ ( x - ) - ( / « - ) l o g , ( x - ) + / M - l = Dat / = l o g , ( x - ) , x € te[-l;l Ta thu duac phuang trinh m = f(t) = Ta CO fit) = 4/^-4 ( / ^ - / + 1) /'-5/ + -;/'(0 = < » / = ±l Ta CO /(1) = -3, / ( - I ) = — Lap bang bien thien cua /(0 tren doan [ - ; l ] , ta thSy / ( r ) lien tuc va nghich bien tren doan [ - ; nen m e -4 thoa man dh bai Xet phuang trinh 4(jc - 2)[log2 (x - 3) + logj(x - 2)] = 15(x +1) Dieukien x > Phuang trinh da cho c6 the viet lai la log2(x-3) + log3(x-2) = 15 x + \ x-2 Ta xet dao ham cua cac ham so tuang ung a m6i ve cua phuang trinh (log, (X - 3) + log3(x - 2))' = I (x-2y x-2 -3 x +1 I + (x-3)ln2 va > (x-2)ln3 < vai x > Suy ve trai la ham so dong bien theo x , ve phai la ham nghich bien theo x nen phuang trinh da cho c6 khong qua mot nghiem Thay x = 11 vao phuang trinh thi thay thoa man Vay phuang trinh da cho c6 nghiem nhat la x = 11 276 Xet bit phuomg trinh log, | l + 2ylx^ - x + 2J + log, (x^ - x + ? ) < Dat t = yjx^-x + 2>0,taCO x^-x + = t^+5 Do bit phuong trinh da cho tra logj (1 + 2t) + log, (t^+5) 2/ ' ' +— > thi day la ham dong bien (l + 201n5 ( / ' + ) l n ^ Ngoai ra, ta cung c6 / ( ) = nen / ( r ) < vai moi t Dieu kien < ^ ^ y-2x-\ [ - < x < Ta thay phuomg trinh 3"" - x - l = c6 dang Bernoulli nen c6 dung nghiem la va 1, ngoai ta cung c6 ' ' - x - l < tren (0;1) va ' ' - x - l > tren (-c«;0), (!;+«)) v'.u' u Ta CO ln(5 + x) - ln(5 - x) = « l n ( + x) = ln(5 -x)5 + x = - x o x = Ngoai ra, ta ciing c6 ln(5 + x ) - l n ( - x ) > vai x > va ln(5 + x ) - l n ( - x ) < vai x < Tu cac dieu tren, ta thay nghiem ciia bat phuomg trinh da cho la x e (1;5) Xet phuomg trinh Vx + sVT^ = log, ((3 - x / (2x +1) Di^u kien < x < Truac hk, ta se chung minh rSng vai moi / e [0;1] thi />log3 (2/^+1) That vay, xet ham s6 / ( O = 3' - 2/^ - , / e [0;1] thi /'(O = 3'In - / , / • ( / ) = 3'(In ) ' - Do / < nen /"(t) < 3(ln 3)^ - < va phuomg trinh / ( / ) - c6 khong qua nghiem Ta cung c6 /(O) = /(1) = nen phuomg trinh / ( / ) = c6 dung nghiem Xet d i u cua hiku thiic / ( / ) tren [0;1], ta c6 3' > 2/^ + « / > logj (2/^ +1) vai moi /G[0;1] T u d o s u y r a Vx + 3Vr^>log3(2x + l) + 31og3(3-2x) = log3((3-2x)'(2x + l ) 277 Dang thuc phai xay ra, tuc la x = 0, x = Vay phuong trinh da cho c6 nghiem la x = 0, x = _ V i du tong hgfp s r * \ Trong phdn nay, ta se xem xet mot so bai tap tong hap ve phuong trinh, bat phuong trinh, he phuong trinh mil lien quan den cdc dang Todn da neu Vi du Giai phuong trinh sau +A" Xog^ x = 0, _ L&i giai *) Dieu kien A: > Ta thay neu x > thi logj x > va 4"^ > nen ve trai duong, khong thoa man Ta chi cin xet < X < Xet ham so / ( x ) = + 4Mog2 x, x e (0; 1) thi / ' ( x ) = 4"ln41og2X + - 4- xln2 Ta can chiing minh / ' ( x ) > 0, Vx e (0; 1) 4" r n ln2 ln4-lnx + — X In • In x + - > 0, Vx e (0; 1) ' ' ^ V Dat — = >'>! thi dua bat dang thuc tren ve ln4-ln —+ >;>0-^>ln4 X y \ny = 0>' = ^ nen khao sat Dat g{y) = - ^ vai ;;>1 thi g\y)=^^—-,g'(>') ln>' ln>' ham so tren mien (l;+oo), ta c6 g{y) > g(e) = e D l thiy e > nen > 2^ = => e > ln4 Do > ln4,V>' > ln_y Dodo / ' ( x ) > 0,Vx€ (0;1) nen ham so / ( x ) da neu dong bien tren (0;l) Tacungco / = nen phuong trinh / ( x ) = c6 nghiem nhat la x = - Vi du Tim s6 nghiem cua phuong trinh sau theo tham so k: (l-x)ln = 2kx-2k + \ L&igidL Ta xet cac truong hop sau: - Neu k = thi thay vao phuong trinh, ta dugc (1 - x) In - = 0, v6 nghiem - Vdri ^ 9t 0, dat - x) = / thi ta dugc phuong trinh / In e Xet ham s6 f{t) = t I n p y , ^ ^ -'- kien-1 < / (DiDH LaigiaL khoi A 2007) a) Xet hk phuong trinh log, (log3(9' - 72)) < ro0 0l < » X > lOgg 73 Ta CO log3(9^ - 72) < X 9"^ - 72 < \ D a t / = 3^>0 nen ^ ' - / - < < » ( / + ) ( / - ) < « / < Suy 3"^ < « X < Do nghiem cua bat phuong trinh da cho la log, 73 < x < b) Xet bat phuong trinh (log, + log4 Bdt phuong trinh tuong duong vod + log2X ^ l o g j x > < ^ loggX ) logj ^llx > Dieu kien < x 9^ l + log2X (log2X + l)>0 log2X Dat / = logj X ?t thi - + t (/ + ) > « Neu / > thi logj x > ^3 + /^^ r/(/+i)>o (/ + l ) > « - ^ />0 / Ngu t\ u(l;+co) Chung minh rSng phuong trinh log' y + log^ X + log^ X log^ CO nghiem va dat la (x,, y^), (xj, - log, y log, x - ( l o g ^ + log^ x) + 80 = ) thi x, + Xj + + ^2 > (Tuyen tap 45 nam tap chi THTT) 279 L&i gidL *)Bihukienxac d\r)h x,y>0 Dat u = log„ y,v = log^ x ta dugc u'+v'+ 3M'V' - 8MV - 6(M^ + M'+V' MV = =8 (W-V)' ) + 80 = (M' + V ' - 8)^ + (MV - 4)' = =0 wv = Giai he nay, ta thu dugc w = v = ±2, tir ta dugc x^ +y^+X2+y2 =a +b + —+ —>4 a Vi du Giai bat phuorng trinh log, yl2x^-3x + \, (x +1) • L&i gidL *) Dieu kien xac dinh 0 - x log, (x +1) > X +1 + l < » 0< j c < - va 0 K6t hgp vai truong hgp dang xet, ta c6 x > 3^ xe 0;- u 1• — u(5;+c») l 2j V 2) Vi du Tim ik ca cac bg s6 thuc (x,>') thoa man d6ng thai ^P-2H->og,5 4^ _ ^^-(,,4) +(^4-3)^ nen 5-(y-4) > 3-iog,5 ^ 5-iog,3 = 5-' ^ + 4) > _ i ^ ^ < _3 Dodo H - | > ' - +{y + 2f =-Ay + {y-\) + {y + 3,f =-Zy-\ {y + 7>f > - l + = =0 DSng thuc phai xay ra, turc la x =-l,x=3 y = -7> y^-3 Vay CO tat ca bo thoa man la {x;y) = (-l;-3),(3;-3) Vi du Giai phuong trinh log^ (11-3^) = log, L&igidL *) Dieu kien {1-2') 11-3^ > 7-2'>0 • Dat y = log3(7-2")r:>2" +3^ = ^ < nen; log,(ll-3^)3^>7^x>- Xet ham so f(x) = l o g ^ O l - ^ ) - l o g ( - " ) , - < x < Ta c6 f'ix) = Do X > —: -3Mn3 2Mn2 -3" T —— + ^ < + ( l l - ^ ) l n ( - ^ ) l n 11-3^ 7-2^ ll-2'-7-3' (ll-3^)(7-2^) 11 ^3 ^ > —^11.2^-7.3^ < , s u y r a / ' ( x ) < , V x e - ; v2y V2 Ham so nghich bien nen c6 khong qua mot nghiem Dong thoi / ( ) = nen x = la nghiem nh4t cua phuong trinh / ( x ) = Vay phuong trinh da cho c6 nghiem nhit la x = Bai tap tong hgp Chuang Bai Giai phuong trinh sau (x +1) log 4' = xlog(x + 2'*^) > Bai a) Chung minh rang moi phuong trinh sau day c6 nghiem nhat CQSX = X (1) sin(cos x) = X (2) cos(sinx) = x (3) b) Goi a,p,/ ' Ian lugt la nghiem cua cac phuong trinh (1), (2) ''3) Chung minh rang/ia f t < y^;''In or < c!r>9 In 281 Bai Giai bat phuomg trinh sau ^ ^ " § ^ jc-l ^— ^Q 2x-\ Bai T i m tat ca gia t r i a de phuomg trinh log^ilS'' -log^ a) = x c6 nghiem nhat Bai Tim m de phuong trinh 4(log2 Vx)^ - log, x + w = c6 nghiem thuoc (0; 1) Bai Chung minh rang cac bo ix,y,z) thoa man J'"'"'"-^"'"^ [xyz=30 ^^,c6it nh4t bo thoa man dSng thuc logj jclogj jylcg, z = Bai Giai phuomg trinh sau log2x^.4 + log3^,^, + log^^,^^, = , ^ • + ^log2 (x' + 2) log3 (x' + 3) log, (x' + 5) Bai Giai phuomg trinh sau ^-s'^^'^^J _^iog,i5 ^ Huang dan giai bai tap tong hgfp Chuong Bai Phuomg trinh (x +1) log 4"^ = x log(x + 2"^^) tuong duong voi xlog4^"'=xlog(x + ^ ^ ' ) « x = v ^ " ' = x + 2^"' Ta xet phuomg trinh 4'^' = ' ^ ' + x , d a t x + l = :>;,tac6 ^ - ^ - > ' + ! = Xethams6 / ( ; ^ ) = 4^ - ^ + l Ta c6 f'{y) = 4nn4-2'\n2-\ De thay rang phuomg trinh f'(y) = c6 dung mot nghiem k va f'(y) duong trenmi^n ik,+oo) vaamtrenmiSn {-00; k) / - , \3 = n n - M n - l = ln2(22 - ^ - , Ta Cling thay rang / ' — > / ' ,4 /1 > 10 m a l n > l o g > — va ^ - ^ > — - = — h a y / ' = ln2(22 - ^ ) - l > ^ 10 3 Do A: < — Ta ciing c6 /(>>) = ' - i + — y=^ f(y)>0,yy o Ro rang A: l a d i l m cue tieu cuaham s6 nen f(y)> f(k)>0 Suy phuomg trinh 4^ - 2^ - >' +1 = v6 nghiem Vay phuomg trinh da cho c6 nghiem nhat la x = 282 Bai a) Xet ham s6 tuong ung voi phuong trinh (1) la / ( x ) = x - cos x Do - < cosjc < nen ta chi can xet x e Taco / ' ( x ) = l + sinx>0,Vxe 2'2 2'2, nen day la ham dong bien Hon nua, / ( ) < , / ( l ) > va day la ham lien tuc nen phuong trinh / ( x ) = CO nghiem nh4t Ta c6 diSu phai chung minh Cac phuong trinh (2) va (3) dugc chung minh tuong tu b) De thay or, /?, 7^ G (0, l ) Bat dang thuc can chung minh tuong duong voi In y3 In Of In / ——< Ta c6: = log3^>-0,5>-l>g(l)>g(x) Suy truong hop bat phuong trinh c6 nghiem la (3) Neu < ;c < 2.Khi bat phuong trinh tuong duong: /(;c) < g{x) Ta thiy / ( x ) > / ( l ) = = g(2)>gW Suy bat phuong trinh v6 nghiem (4) Neu < JC < Khi bat phuong trinh tuong duong: / ( x ) < g{x) Ta c6: f{x) > / ( ) = log3 > , > , = | = g(3) > g(x) Suy /(jc) > g(x) Do do, b i t phuong trinh v6 nghiem (5) Nea Khi bat phuong trinh tuong duong: / ( x ) < g(x) Ta c6: f(x) > / ( ) = > g(x) nen suy r a / ( x ) > gix) Do do, bat phuong trinh v6 nghiem T u cac truong hop tren, ta c6 nghiem cua b i t phuong trinh la ^ < x < Bai Phuong trinh da cho tuong duong voi 25^-log5a = 5^ ' ^ - ^ - l o g , f l = « ''^^ [t'-t-\og,a =0 (*) Phuong trinh da cho c6 nghiem nhit (*) c6 dung nghiem duong hay -t = log; a CO dung nghiem duong Xet ham s6 f{t) = t^-t Taco: f\t) = 2t-\ voi t e [0; +oo) /'(0 = 0«/=i; /W = - ' / ( ) = 0- \2) Dua vao bang bien thien, ta suy phuong trinh / ( / ) = log5a c6 dung nghiem duong a>\ log5a>0 log.a^ a= Bai Dieu kien jc> Phuong trinh da cho tuomg duong vai log]X + logjx + m = 0; xe(0;\) Dat / = logj X Vi lim log^ x = -oo va lim log x = 0, nen xe(0;l)=>/e(-oo;0) « Ta c6: -t-m = 0, t — - — vai a,b,c>\ •+ T> \ a \ b ; l + c ^ l>+ abc + fl' l + b^~l + ab (l + a^)il + b^) l + ab thong qua bat dang (2 +thuc a'—+ ^6'sau)(1 + —+ ab) ^ >> ^2(1—+ a^+b^+ vai a,b>la^b^)(*) l + a' ++6'ab{a^l ++b^)>2 ab ^2 + a^+b^+2ab + 2a' +2b^ + 2aW That vay, 0 Bit ding thuc cu6i dung1 nen (*) dugrc chung + a ' +minh 6' Do 1 1 ^ 2 ^ ^ - ; = > •l + abc l + a^• + + 6'r + l + c'r + l + abc > l + ; + l + c^yfab 285 nensuy r- + \ a' \ b' Tir suy log,, r- + l + c' r-> l + abc vm a,b,c>\ + W , + log,, , 5> + ^logj ix'+2) log3 (x' + 3) log3 {x' + 5) Dang thuc xay x = Vay phuong trinh da cho c6 nghiem la x = Bai Dieu kien < X Dat = a > 0, ta CO ' - - x'""^'' = x'"^'' • x = ox va Q Q Q Q Q\ Q\ Q ' a X = — Tir taCOphuorngtrinh x^-ax +—a^ = ( x - a ) ( x - a ) = • » 2a x= — Ta CO truong hop: - Neu ^ = ~ thi ta dugc X =• logj X = logj • log3 X -1 o log3 X (1 - log3 5) = -1 108,3 logj X = logj X = ' jr - Neu X = ^ X= thi ta dugc 2x'°^'' ? r 2 « logj X = logj • logj X + log3 - » log3 x(l - logj 5) = log, - '°^'3 » logj X = l0g3 - X = ' log, logj? _ , - Vay phuong trinh da cho c6 nghiem la x = ' , x = ' 286 QYaiHag P H l / d N G T R I N H , BAT PHlTCfNG T R I N H HtTU T I §1 Tam thufc, phi/cfng t r i n h , bfit phiTcfng t r i n h bac hai §2 Phtfcfng t r i n h , bat phifcfng t r i n h bSc cao phiicfng t r i n h , bat phifcfng t r i n h hOfU t i V i du tdng hgp Chifcfng 31 66 Chtftfng PHl/CfNG T R I N H , BAT P H l f d N G T R I N H V T I A Mot s6' dang phiicfng t r i n h co ban 80 B Phifcfng phdp giai 1) C^c phifcfng phdp dai so 2) Cdc phiicfng phdp liicmg gidc 82 gidi t i c h 3) V i du tdng hop Chi/cfng Biii tap tong hcrp Chiicfng 135 167 177 Chtfdng PHLfCfNG T R I N H , B A T P H l / d N G T R I N H MU A T6m tat ly thuyet 182 B Phirong phap giai 1) Phiicfng phap bieh ddi ve cung ccf s6' hoac lay logarit hai ve 184 2) Phtfcfng phap dat ^ n phu 190 3) Phiicfng phap phan tlch 197 4) Phiicfng phap danh gia 205 5) Phiicfng phap dung ham so 212 6) V i d u t n g hcfp 224 Bai tap tong hcfp Chiicfng 231 Chifrfng PHlTCft^G T R I N H , B A T PHtfdNG T R I N H L O G A R I T A T6m tat ly thuyg't 245 B Phifcfng phap giai 246 Phiicfng phap bien doi ve cung ccf s6' 246 Phiicfng phap dat an phu diTa ve dang dai so 251 Phi/cfng phap dat an phu di/a ve dang mu 258 PhiTcfng phap danh gia 264 Phiicfng phap ham so dcfn dieu 271 V i du tdng hcfp 278 287 y • y / / y y y / y y- y y ^ y • X \ \ \ Email: nhasachhongan@hotmail.com C N g u y e n Thi M i n h Khai - Q.1 - T P H C M DT: - 7 - 9 • F a x : • \ \ A (^^SK- ^amy ti^rri' e^bo: - 245 Tran N g u y e n Han - HP * DT: 3858699 - & P h a n B p i C h a u - H^i P h o n g *DT: - 04 Ly Thai To - TP Da - 259 Le Duan N i n g - TP Vinh *DT: - DT: - 15 Le T h a i T o - VTnh L o n g - DT: 3823421 ^ -< ISBN: 978-604-62-2269-9 \ 3554777 - 39-41 V Thi Sau - Can T h d * DT: • TTnh 1^ - T T C u C h i - T P H C M 3839599 3818891 *DT: \ 37924216 0907845219 \ 935092 763842 G i a : TO.OOOd ... g TS LE XUAN S d N (CB) • ThS PHAN V I E T BAC ThS TRAN NHAN - C N LE PHUC L Q HUUTI VO Tl MO LOGARIT ^ D U N G CHO HS GIOI, THI TRI/flfNG CHUYEN y ^ O N THI THPT QUOC GIA DTR: N H A X U A T... ban: NHA SACH HONG A N SACH L I E N K E T PHaONG TRINH - BAT PHUONG TRINH HLJfU TJ, VO TJ, MU, LOGARIT Ma s5: 1L - 99OH2015 In 2.000 cudn, kh6 17 x 24cm tgi COng ti Cd phin VSn h6a VSn Lang Dia... phuang trinh v6 ty; ChiroTig va Chirong theo thu tu de cap den phuang trinh, bat phuang trinh mu va logarit Trong tung muc, tung phuang phap deu c6 cac vi du minh hoa tieu bieu; c6 phan bai tap de