ChUcfng PHtfCfNG TRINH, BAT PHlJdNG TRINH MU A Tom tit ly thuylt Cac cong thijfc bien doi dai so: Duai day la cac cong thiic h'lkn d6i ca ban \h luy thua, ap dung de bien doi cac phuang trinh mu: , in oam,^ < a a' a a'"-b'"=iabr,^ = vai a , > , m E ' I— a"* = _ , ^ a ' " = a " vai a > 0,6 e ! , / « , « e Z \ a Cac npi dung ve giai tick: • • - Tap xac dinh va tap gia tri: Ham so luy thua y = x'' c6 tap xac dinh nhu sau: Neu a e thi x € M, neu a e Z " u { } thi X G ] R \ { } , ngu a Z thi x e ,^ vox Q < a ^ \i tap xac dinh la R va tap gia tri la M"" Dao ham cua cac ham s6 (x") = x""',{a'')' = a'\na • Ham s6 mvi y ^ - V a i a>\, - Giai han cua ham so: • N6u < a < thi cac ham s6 y = a'' nghich bi6n - Neu a > thi cac ham SO - Tinh dan dieu cua ham s6: • , = a"" dong bien tac6 cac giai han sau lim a" = +oo, lim a" = JC->+-00 V o i < a < ta CO cac giai han sau lim a" = 0, lim a"" = +oo 182 Cdc dgingphirfftig trlnh ca ban: 5^,;; ; -n, , ? / :i) • Phuongtrinh a" =m vai ni>0, luonc6nghiemduy nhat la x = log„m • Phuong trinh a^^"^ = a^'-"^ / ( x ) = g{x) tinh don dieu cua ham s6 mii • Phuong trinh a^*""' =ft^*""^voi a^b thi tinh logarit ca s6 a true tiSp hai ve, ta duge - ~ log4a^^^') = log4M)«/(x) = g(x)-Iog,6 - Day la cae dang eo ban va noi ehung hau het cae phuong trinh mu va logarit tir ea ban den nang cao deu phai thong qua Trong goc nhin nay, eon mot so dang bien nhu phat hien tinh doi xung de chia va dat §n phu, biSn doi ap dung eong thuc thich hop de dua ve eung ca so, Cdc dfing bat phuang trlnh mu ca ban: Tuang ung voi cae dang cua phuong trinh, bat phuong trinh ciing eo eac dang cabannhusau: • Dang a'' > vai < o 7i : ta c6 truang hop nhu sau: " - N6u < thi bit phuong trinh nghiem dung vai moi x - Ngu > thi ta CO x> log„h neu a > va A:< log„h neu a 0, x /3x+2> 12X+8 J = 2"3 2'°^ « l Phuong trinh da cho tuong duong voi n = 2>°^ « 2"^^3 , o ^ ^ + ^ = 1OV^ x-1 Khu mau va nit ggn, ta dugc 4x + = 3(x - l)\/x 184 Dat/ = 0,taCO 't = 3/^+2/ + l = ' De thay phiromg trinh thu hai v6 nghiem nen phuang trinh tren c6 nghiem la t = Vai t = thi X = nen phuang trinh tren c6 nghiem la x = Vi du Giai phuang trinh 2"' = ' ' L&igidL Ta c6 4-2' 2'*' = 16'' 2'" = T'' _ ^2-" -.2"^ « 2'" = ' " « •>4x 2'^ = T2X + 4x = x + 2jc = - Vay phuang trinh da cho c6 nghiem nhat la ^ = " j • 2x-3 V i du Giai phuang trinh 3^"' • ^ = 18 LoigidL *) Dieu kien xac dinh jc 9^ o Ta c6 2x-3 3^^-'-4^ 4x-6 =18«3^'-'-2^ 6-3x = - ' « ^ ' - " =2" Lay logarit ca so hai ve, ta dugc x'-4 , = 6-3x log, « ( x - 2)(x + 2)x + 3(x - 2) logj = + 2) + log3 2] = (x - 2) x-2 x=2 =0 x(x + 2) + 31og32 = (x + 1)' +31og3 - l = D I thiy phuang trinh thu hai v6 nghiem nen phuang trinh ban dau c6 nghiem la X = V i du Giai phuang trinh sau 2^"' • 3" = 4^' • ^ L&igidL *) Dieu kien xac dinh: x^-l 185 2x Taco 2^^'-3'=4^ • 36^^' o • 6" = ' •6^^'o4'-^ =6'^' Lay logarit ca so hai v l , ta duQfc x+1 "l-x = , log4 « (1 - x) [(x +1) - X log4 = x=l (x +1)' - x l o g ^ = " [ x ' + x(2 -log4 6) + = D I th4y phuomg trinh thu hai v6 nghiem nen phuomg trinh ban dhu c6 nghiem la x = l V i du Giai phuong trinh sau ( ^ " ' + ( s ^ = [l"''- • (2"* J Z,^i^/fl£ Phuomg trinh da cho tuomg duomg voi O 26 • 5^ =26-r « 5' =r o (x' + 3x) log^ = x' + 6x « x[(x + 3) log2 - (x + 6)] = "x = 125 logjS-l 64 6-31og,5 , ' i^^ - • Vay phuomg trinh da cho c6 hai nghiem phan biet la x = 0,x = log 64 125 V i du Giai bdt phuomg trinh sau 3^^''^' > (DH Bach khoa Ha Noi 1997) L&i gidL * ) DiSu kien xac dinh: x^ - 2x > / I X > J- , x | x - l | < x < = > - x < x - l < luon dung ^ , > - thi bat phuomg trinh (*) _ 186 NSu \x-\\-x>0^x - 4x + >\-2x, - 2x +1 < 0, v6 nghiem Do do, nghiem cua bat phuong trinh da cho la x > : 4" +2x-4 Vf du Giai bat phuong trinh sau < x-1 (DH Van Hoa Ha Noi 1997) Led giai *) Dieu kien xac dinh x*\ Ta c6 x-\ Ta CO truong hcfp: - Neu -Neu < 4^-2l 4^-2>0 f4^>2 x-l2' -r x>0 3^-3-2^ > 3^-2^ < 2) =3 /=3 'x = \ x =0 Vay phuofng trinh da cho c6 nghiem la x = 0, x = x-^ jc-3 3x+l Vi dy Giai phuomg trinh 8^*^ + 2^^^^ = _ L^/^/ai *) Dieu kien jf i - Ta CO - ^ ^+2+2 ^"^2 o4 x+2 la du Xet ham so / ( x ) = log3(7''+2)-log5(6''+19) lien tuc tren [0;+oo), taco: fix) = (6"+19)ln5' (7^+2)ln3 6Mn6 7Mn7 Ta cung CO I n > 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 \ 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 [...]... 2" ^ Taphan tich nhu sau : • - -^, • (2' " - 2 ' " / ) + (2' "/ -2' "e) + ( 2 ' " t - 2 ' " ) + [t' -2U^)-^[t' » 2' " ( 2 ' -t) + 2' 't (2' -1) (2" +t)- 2' " (2" -t)-t' /3+l) log2 X = 0 «> X = 1 =0 ma'! Vay phuang trinh da cho co nghiem duy nhat la x = 1 Vi du 4 Giai phuang trinh 4" - 2" ^' + 2( 2" -1) sin (2" + -1) + 2 = 0 (Du bi khSi D 20 06) L&igidL Phuang trinh da cho tuang duomg voi (2' " - 2. 2" +1) + 2( 2" -1) sin (2" + -1) +1 = 0 o (2" - 1 ) ' + 2( 2" -1) sin (2" + > ' - ! ) + sin' (2" + 2" - 1 + sin (2" +y-1)J' -l2 -1) + cos' (2" + -1)... 3^ + 3^"' = 2 x ' - 3 " + 2 x + 6 Ta bien doi nhu sau x ' ( 4 - 2 - 3 ^ ) + x ( 3 ^ - 2 ) + 3^"'-6 = 0 o - 2 x ' (3" - 2) + x(3^ - 2) + 3(3^ - 2) = 0 (3^ - 2) (2x' - x - 3) = 0 3^ = 2 2x'-x-3 = 0 X = logj 2 , 3 x = -l,x = - Vay phuang trinh da cho c6 3 nghiem la x = logj 2, x = - 1 , x = — 2 Xet phuang trinh 5 = 625 • 125 " + 1 Ta b i l n d6i nhu sau -2x'-3x +2 2x'-3x + 2 = 0 y+3x +2= 0 -1 = 0... -4 < m < - 2 Vi du 5 Giai phuomg trinh 4 ^ ^ ' + 2 2 ^ " =2- 4" L&igidL Phuomg trinh da cho tuong ducmg voi 4 2^ ^^^^ + 2 • 2^ "^'^" - 2 • 2' " - 0 o 4 • 2' Dat / = 2^ '^'-" > 0 , ta dugc 4 r + 2 / - 2 = 0 + 2. 2-"^'-' - 2 = 0 2 / ' = "/ = - 1 0 « 1 • t= 2 Ta chi nhan nghiem r = ^ va khi do 2^ '-' =- 0 r+ x-ll 0 Xet ham s6 / ( x ) = 2" + x -11 thi / ' ( x ) = 2Mn 2 +1 > 0 nen no dong biSn tren R Mat khac, / ( 3 ) = 0 nen ta c6 / ( x ) > 0 « x > 3 va / ( x ) < 0 » x < 3 Xet ham s6 g(x) = 2 - 2 " - x ' - x - 2. .. ra khi jc = 2 Thu lai ta thay thoa man Vay phuong trinh da cho c6 nghiem duy nhat la x ^2 ( Vi du 7 Giai phuong trinh sau M( 2^ +4^ V J = 144 L&igidi *) Dieu kien: x ^0 20 8 T ( Chu y rang vai x < 0 thi M 2V2^ -2^ =2- 22^ ^ >2- 2 ^^'^ =8 va =3^+3 ^ >2V3^-3^ =2- 3^^^ >2- 3 ^^'^ =2- 3^ =18 ^\ ( 2^ 3^+9^ >8-18 = 144 \ V ) 4 X 2 Dau bang... V 3 N/3 ( 2 - V 3 ) + (2- V3p"-' (2- V^) = 4 + (2 - V 3 = 4 > 0 thi 1 = (2 - > / 3 v a ta dugc t + - = 4t^-4t + \ 0^ t - V o i / = 2- 73 « ( 2 + >/3r'-'^ =2- 73 » x ' - 2 x = - l « x = l - V a i t = 2 + S0 Trong tnrong hop nay, phuang trinh (*) a tren khong dugc thoa man - Neu cos 2JC < 0, lap luan tuang tu truomg hgp tren ' ^ - N8u COS2JC = 0 thi (*) dugc thoa man va... - x ' < 24 - X[1 - ( x ' - 9) • 2^ "" LcigiaL Dat r = 2" ,r > 0, bdt phuang trinh da cho tra thanh: , 10x' +22 2t + 2 ^/ ^'-9x x' ... t = -x^ + - X 2 / = -x + ll , +1 Do do, bat phuong trinh da cho tuong duong: (2/ -x^-x -2 ) (/ + x-ll) 0 r+ x-ll0... = 2" ^ Taphan tich nhu sau : • - -^ , • (2' " - ' " / ) + (2' " / -2 '"e) + ( ' " t - ' " ) + [t' -2 U^ )-^ [t' » 2' " ( ' -t) + 2' 't (2' -1 ) (2" +t )- 2' " (2" -t)-t'