88664422-22-1 01-55551 01ggx ( ))= 22xxffx ( ))= 2GII TCH 12PHN 2: Nm hc: 2010 - 2011LY THA11.NH NGHA LY THA V CN.S m C s aLy tha a*N n R a n a a a a an( ...... . tha s )0 0 a10 a a) (*N n n 0 annaa a1 ) , (*N n Z mnm 0 > a) ( a b b a a a an n n mnm ) , ( lim*N n Q r rn n 0 > a nra a lim 2. TNH CHT CA LY THA.* vi a > 0, b > 0, ta ca.a .a a ; a ; (a ) a ;aa a(ab) a .b ;b b + _ ,a > 1 : > > a a0 < a < 1 : < > a aBi 1:n gin biu thc.1) ( )552312 6. . y x y x 2)3 33434b aab b a++3)1 .1.14142134++++aaa aa aa4)

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+++mmmmm1212.2 2421322Bi 2:Bin i a v dng ly tha vi s m hu t.1)7 3 5. 281ax2)3 4 5. a a3)4 8 3. b b4)4 3. 2731aBi 3 :Tnh .1)( )333

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2)3 1 3 2 116 . 4+ 3)2 323274)( )55482Bi 4:n gin cc biu thc.1)1) (2 3 23 2 2 2+b ab a2)3 3 43 3 3 3 2 3 2) )( 1 (a aa a a a+ + 3)

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+ ab b a . 4 ) (124)4 1 23 3 31 3 14 4 4a a aAa a a _+ , _+ , 5)1 1 12 2 21 12 22 2 112 1a a aAaa a a _+ + + + ,6)1 7 1 53 3 3 31 4 2 13 3 3 3a a a aAa a a a + 7)1 11 12 24 43 1 1 1 14 2 4 4 4:a b a bA a ba a b a b 1 _ 1 1 ,+ + 1 ]38)1 1 1 112 4 2 21 1 14 4 21 1 21 1x x x x xAx x x 1 1 + + +1 1 + 1 1 1 + 1 ] ]Bi 5: Rt gn:a) ( ) 1 1 _ 1 1 _ 1 ,+ 1 , ]1122 3 31 12 22 21 a bA aba ba bb) +2 21 1 3 1 12 2 2 2 2a a 2 1 aBa a a a ac) 2 2 112 1a a aCaa a a _ _+ +

+ + , ,d) ( ) ( )( )12 3 4 3 312 3 3 3 31 a a aDa a a + +e)2 8 5 13 3 3 32 5 2 13 3 3 3a a a aEa a a a + Luyn tp1/. Vi t di dang luy tha vi s mu hu ti ca c biu thc sau :a/. 5 32 2 2 b/. 116: a a a a a; a > 0. c/. 2 4 3x x; (x > 0) d/. 5 3a ab b; (ab > 0)2/. n gian cac biu thc sau :a/. 4( 5) a b/. 4 281 ; ( 0) a b b 1 ]g/.h/. 3 5 13 48 + +3/. a nhn t ngoai vao du cn :a/.(4 ) ; ( 4)4xx xx >b/21(5 ) ; (0 5)25a aa < > c/. 13 2 + d/. 54 11 +e/. 3 315 2 5/. Tinh gia tri cua biu thc :a/. 15 1 3 7 1 123 3 2 4 4 23 .5 : 2 : 16: (5 .2 .3 A 11 ' ; 11 ] ] b/. 2 3 333 2 2223:( )a b a a bAa a b ba ab+ ; vi 65a va 35b c/. 32 3 12 13 2 2( ) ( ) A a b ab a 11 ]; vi 22a va 312b 56/. Chng minh ng thc sau : a/. 1 2 221 1 1 1 32 2 2 2 21 20a a aaa a a a a + + +b/. 3 3 3 3 2 4 2 2 2 4 2 2 3( ) a a b b a b a b + + + +c/. 3 2 2 3 2 2 2 + d/. 3 35 2 7 5 2 7 2 + 7/. Rut gon biu thc :a/. 12 2 1.( )aab/. 23 ( 3 1): b bc/. 42 4: x x x d/. 3 325 5( ) a8/. So sanh a/. 6003 va 4005b/. 571( )2 va 3142.2c/. 33va 2HM S LY THAI.Khi nim:Hm sy x ; , c gi l hm ly thaCh : tp xc nh ca hm s ly tha ph thuc vo gi tr ca - Vi nguyn dng th tp xc nh l R- Vi nguyn m hoc bng 0, tp xc nh l { }\ 0 - Vi khng nguyn th tp xc nh l( )0; +Lm bi 1/ 60II. o hm ca hm s ly tha:( ) ( )1 1x ' .x ; u ' .u Lm bi 2/61LOGARITI. Khi nim logarit 61. nh ngha:Cho 2 s a, b dng vi a khc 1. S tha mn nng thc a bc gi l logarit c s a ca b v k hiu logab( ) 1 log b a ba V d 1:Tm x a) log 42x b)2log 3 x c) 811log4x d) log 25 2xb) e) log ( 1) 23x + f) ( )log 432 4 x g) log ) 4 (212x h) log 13 4152x _ ,k) log 5) 0 (42x + l) log 2 8xCh : khng c logarit ca s 0 v s m2. Tnh cht: ( )( )( )( )( )2 log 1 0a3 log a 1alog ba4 a b5 log aaV d 2: Tnha) log 324b) 433logc)322logd) 2log 4e) 31log3f) 21log16g) 132aalog( )vi0 1 a < h) 3 57 4949+log logi) 1 13 26 89 4 +log logII. Quy tc tnh logarit :1. Logarit ca mt tch : a > 0; b1> 0; b2> 0, a 1 7( )( )6 log b .b log b log ba a a1 2 1 2 +Logarit ca mt tch bng tng cc logaritV d 3: Tnh:a) 12 12log 6 log 2 +b) 1 1 12 2 24log 6 log 24 log9+ +2. Logarit ca mt thng: a > 0; b1> 0; b2> 0, a 1 ( )2b17 log log b log ba a a1 2b _ ,Logarit ca mt thngbng hiucc logarit( )18 log log ba ab_

,V d 4: Tnha) 100 425 25 log log.b) 2 2 220 6 15 log log log + .c) 2 2 25 10 25 log log log + .d) 6 7 143 3 3log log log + e) 10 7 145 5 5log log log + .3. Logarit ca mt ly tha : a > 0; b> 0, a 1 ( ) ( )9 log b log ba aLogarit ca mt ly tha bng tch ca s mvi logarit ca c s( )( )n110 log b log ba an V d 5:Cho log 2; log 3 b ca a . Hy tnh log xa,bita)2 34a bxc b) 23a bxc c) 2 2 3x a bc III. i c s : Cho a > 0; b > 0. c>0, a 1 ,c 1 8( )log bc11 log balog ac( )112 log balog ab b 1 ( )113 log b log baa;0 V d 6: a) Cho5 142 2log ;log a b . Tnh 352log theo a v bb) Cho 10 72 2log ;log a b . Tnh 352log theo a v bc) Cho4 53 3log ;log a b . Tnh 103log theo a v bd) Cho 2 95 5log ;log a b . Tnh 65log theo a v be) Cho 3 5 27 2 3log ;log ;log a b c . Tnh 5063logIV. Logarit thp phn, logarit t nhin1. Logarit thp phn: l logarit c s 10log b10thng vit l logb hay lgb2. Logarit t nhin:l logarit c s elog be thng vit l lnb Ch : logblog baloga lnblog balnaLuyn tp:Bi 1: Bit log52 = a v log53 = b . Tnh cc lgarit sau theo a v b.1)log527 2) log5153)log512 4)log530Bi 2: Lgarit theo c s 3 ca mi biu thc sau , ri vit di dng tng hoc hiu cc lgarit.1)( )325 3b a2)2 , 06 510

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ba3)549 b a4) 7227ab 9Bi 3: Tnh gi tr cc biu thc.1)log915 + log918 log9102)331313145 log 3 400 log216 log 2 + 3)3 log212 log61 364)) 3 log . 4 (log log2 341Bi 4: Tnh gi tr cc biu thc.1)1 1log 4log 8 log 29125 7 4 281 25 .49 _ + ,2)1log 3 3log 51 log 55 24 216 42+++3)1log 4log 9 log 67 7 5272 49 5+ _ ,Bi 5:Tm x bit.1)log6x = 3log62 + 0,5 log625 2 log63. 2)log4x= 3 log 4 10 log 2 216 log314 4 4+ Bi 6:Tnh.1)20 20) 3 2 log( ) 3 2 log( + + 2) ) 7 2 5 log( ) 1 2 log( 3 + +3)ee1ln ln +4) ) . ln( 4 ln2 1e e e +Bi 7:Tm x bit1) logx18 = 4 2)532 log5 x3) 6 ) 2 . 2 ( log3 xBi 8: 1) Bit log126 = a ,log127 = b. Tnh log27 theo a v b.2) Bit log214 = a. Tnh log4932 theo aHM S M HM S LOGARITI. Hm s m: 1. nh ngha:Cho a 0, a 1 > Hm s y = axc gi l hm s m c s a.102. o hm ca hm s m: ( )( )x xe ' eu ue ' u ' e

( )( )x x' au u' u ' aa .ln aa .lna3. Kho st hm s m xy a , a 1 >xy a , 0 a 1 < Hm s y =logax c gi l hm s logarit c s a2. o hmca s logarit : 11( )( )1log x 'ax. ln alog 'au. ln au 'u( )( )1ln x 'x1ln u ' .u 'u3. Kho st hm s logarit y log x, a 1a > y log x, 0 a 1a < lim ; limy ;x 0 xy+ + +lim ; limy ;x 0 xy+ + +Tim cn ng : trc OyBBTx0+y+y+ -BBT4422-22-44-1 01-55551 04422-22-44-1 01-55551 0Bi 1: Tm tp xc nh ca cc hm s sau.1) y = 1 xxee2) y =11 2 xe3) y = ln ,_

xx11 24) y = log(-x2 2x )125) y = ln(x2 -5x + 6) 6) y =

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+ xx x3 11 3 2log22Bi 2: Tnh o hm ca cc hm s sau.1) y = (x2 -2x + 2).ex2) y = (sinx cosx).e2x3) y = x xx xe ee e+4) y = 2x - xe5) y = ln(x2 + 1) 6) y = xx ln7) y = (1 + lnx)lnx 8) y = 1 ln .2 2+ x x9) y = 3x.log3x 10) y = (2x + 3)e11) y = xx .12) y = 3xBi 3: Chng minh rng mi hm s sau y tha mn h thc tng ng cho.1) y = esinx; ycosx ysinx y = 02) y = ln(cosx) ; ytanx y 1 = 03) y = ln(sinx) ;y +ysinx + tan2x = 04) y = ex.cosx ;2y 2y y = 05) y = ln2x ;x2.y + x. y = 2Bi 4:Cho hm s2x xy e +. Gii phng trnh y y 2y 0 + + Bi 5: Tm gi tr ln nht, gi tr nh nht ca hm s1). xy x e trnon [ 1; 2] 2)+xxeye e trn on [ln2; ln4]3) y = lnx x .4)( )2y x ln 1 2x trn [-2; 0] ( TN08-09)5) y = 22log 2log 2xx+ trn on [8; 32]6) y = f(x) = x2 - 8. lnx trn on [1 ; e]7) f(x) = (x2 3x +1)extrn on [0;3]8)y = x lnx + 3 trn 1; ee 1 1 ]9) f(x) = x2e-x trn on [-1;1]1310) 2ln( )xf xxtrn on [1;e3]PHNG TRNH M PHNG TRNH LOGARITA. PHNG TRNH MI. Phng trnh m c bn( )xa b a 0;a 1 > Nu b > 0 th phng trnh c duy nht mt nghim x log baNu b = 0 hoc b < 0 th phng trnh v nghimV d1: gii cc phng trnh sau:a) x10 1 b) x8 2 c) x4 4 d) x5 e f) x2 3 g) x1327 h) x912 _ ,II. Mt s cch gii phng trnh m1. a v cng c s: 0 a 1 < ( )( )f x ba a f x b ( ) ( )( ) ( )f x gxa a f x g x V d2: gii cc phng trnh sau:a)2x 5x 61 5 +b) . 3x 1133 _ ,c). 2x 3x 24 16 +V d3: gii cc phng trnh sau:a) 2x 2x 31x 177 + _ ,b). 2x 214 3x22 _ ,c) ( )5 x2x 3 40, 753 _ ,d) ( )( )x2 3x0, 5 2+14e) 2x x 8 1 3x4 2 + f) x 112x12525+ _ ,V d4: gii cc phng trnh sau:a) x 1 x 2 x 3 x 43 3 3 3 750+ + + b) 2x 1 2x3 3 108+ c) 2x 1 2x 15 3.5 550+ d) x 1 x 1 x2 2 2 28+ + + e) x 1 x 1 x2.3 3 3 9 6.+ f) 2x 7116 16x x.4 82 _ ,2. t n phDng 1: Phng trnh 2x xA.a B.a C 0 + + Cch gii:t xt a , iu kin: t > 0Gii phng trnh theo t: At2 + Bt + C =0, chn t tha kSuy ra xa t x log ta V d 5: Gii cc phng trnh sau:a)12x x.5 5.5 2505+ b)2x 2 x2 9.2 2 0+ + ( tt nghip nm 2005 2006)c)2x 1 x9.3 6 0 3+ + ( tt nghip nm 2007 2008)d)2x 6 x 72 2 0 17+ + + e)x x.3 0 9 2 15 f)x x0 64 8 56 g)x x.5 0 25 6 5 +( tt nghip nm 2008 2009)h)x x 1.3 0 9 24 15 +i)4x 8 2x 53 4.3 27 0+ + + j)x x 14 36.2 32 0 + k)6x 3xe 3.e 2 l)2 2x 5 x x 5 x 24 2 4+ + + Dng 2: Phng trnh c cha ax v a-x, hoc ax v bxvi a.b =115t:x x1t a a ; t 0t >V d 6: Gii cc phng trnh sau:a)x 1 x3 18.3 29+ + b)x 1 1 x3 3 10+ + c)x 1 x5 5 4 0 + d)2x 2xe 4.e 3 e)2 2sin x cos x9 9 10 + f)2 2sin x cos x4.2 6 2 + g)( ) ( )x x4 15 4 15 62 + + h)( ) ( )x x2 4 2 3 3 + +i)( ) ( )x x6 4 6 35 35 + +Dng 3: Phng trnh 2x x x 2xm.a n.a .b p.b 0 + + Cch gii: Chia 2 v ca phng trnh cho mt trong 3 s 2x x x 2xa ; a .b , b a v dng 1 hoc 2V d 7: Gii cc phng trnh saua)x x x2.25 7.10 5.4 0 + b)x x x5.36 3.16 2.81 + c)x x 2x 125 10 2++ d)x x x0 4.9 12 3.16 + e)x x x3.4 2.6 9 f)1 1 1x x x4 6 9 + g)2x 4 x 2x 23 45.6 9.2 0+ ++ h)x x x3.25 2.49 5.35 + ( Phn 3, 4 ch dnh cho lp 12C1 tham kho)3. Phng php logarit ha S dng tnh cht: 16Nu 0; 0 > > v log log ; 0 a 1a a < Thng s dng phng php ny khi gp phng trnh c dng:( ) ( ) f x gxa bLy logarit cng mt c s a n thot ra khi s m.V d 8: Gii cc phng trnh saua)x 1 x2 .5 200+b)2x 4 x 22 3 c)2x 5x 6 x 35 2 + d)2x 1 x x 23 .2 8.4 e)x x x 15 . 8 100+4. Phng php n iu: Cch gii: Ta ch ra mt vi nghim ca phng trnh ( thng dng ny c duy nht mt nghim). Dng tnh n iu chng minh phng trnh khng cn nghim khc na.Ch :Khi a> 1 th yxx y a a > >Khi 0'>Phng trnh cho tng ng vi: f(x) = g(x)V d 2: Gii cc phng trnh:a)( ) ( )log 5x 3 log 7x 53 3+ +b) ( ) ( )2log x x 7 log x 3 6 + c)( )log x log x 1 12 2+ d) ( ) ( )log x 5 log x 2 32 2 + + e) ( ) ( )log x 1 log 2x 11 log 2 f) ( )log log x 3 22 4x g) ( )log log x 23 3x 1 + + h) ( ) ( )2log x 3 log 6x 10 02 21 +i) ( )22log log x 7522x + j)25log x log x log x log x2 4 8 1612+ + + k) ( ) ( )21 1log x x 5 log 5x log2 5x _+ + ,l)( ) ( ) ( )21log x 4x 1 log 8x log 4x2 m)log x 4log x log x4 8132+ + n)log x log x log x 63 133+ + o)x 8log log xx 1+182. t n ph:V d 3: Gii cc phng trnh:a)( )log x log 4x 54 2+ ( tt nghip nm 2006 2007)b) log23(x+1) 5log3(x+1)+6 = 0c)2 2log ( 1) 3log ( 1) log 32 02 2 2+ + + x xd)+ log 16 log 64 32 2xxe)log 2 2log 4 log 8x2x2x+ BT PHNG TRNH M LOGARITI. Bt phng trnh m: 1. Bt phng trnh m c bn:l bt phng trnh c mt trong cc dngx x x xa b (a b, a b, a b) > < , vi 0 a 1 < gii bt phng trnh m ta s dng tnh n iu ca hm s m, Ta xt bt phng trnh xa b >Nub 0 th bt phng trnh c tp nghim l RNu b > 0 th bt phng trnh tng ng vi log bxaa a >Vi a > 1 th bt phng trnh c nghim x log ba>Vi 0 ,2. Mt s bt phng trnh n gin: c cch gii tng t nh gii phng trnh . Ch n tnh n iu ca hm s mV d 2: Gii cc btphng trnh:a)2x 3x2 4 +19e)2x 1 2x 2 2x 32 2 2 448 + + f)x x2 2 3 0+ h)x x x5.4 2.25 7.10 + II. Bt phng trnh logarit 1. Bt phng trnh logarit c bn: l bt phng trnh c mt trong cc dng sau:( )log x b log x b; log x b; log x ba a a a> < gii bt phng trnh logarit ta s dng tnh n iu ca hms logaritTa xt bt phng trnhlog x ba>,0 a 1 < Vi a > 1 th bt phng trnh c nghim bx a >Vi 0 c) 51log x2d) 2log x 4 < e) log x 13 f) 13log x 2 2. Mt s bt phng trnh n gin: c cch gii tng t nh gii phng trnh . Ch n tnh n iu ca hm s logarit.V d 4: Gii cc btphng trnh:a)( )log 4 2x 28 b)( ) ( )log 3x 5 log x 11 15 5 > +c)( )log x log x 2 log 35 0,2 0,2 s:( ; 2) (4; ) x + DB_B_2004BT PHNG TRNH LOGARIT1.33 5log ( ) 11xx +. s: 2 x < DB_A_20082.( )1 1 22 4log 2log 1 log 6 0 x x + + s:3 x DB_B_20033.224log [log ( 2 )] 0 x x x+ . s:2 3 0 x + < x x x13.22log2log2 20 0xxx + 214.24