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ôn thi đại học chuyên đề logarit và mũ loga ôn thi đại học chuyên đề logarit và mũ loga ôn thi đại học chuyên đề logarit và mũ loga ôn thi đại học chuyên đề logarit và mũ loga ôn thi đại học chuyên đề logarit và mũ loga ôn thi đại học chuyên đề logarit và mũ loga ôn thi đại học chuyên đề logarit và mũ loga
Gia s Thnh c www.daythem.edu.vn PHN I : KIN THC C BN CN NM VNG HS M: y = ax (a > 0, a 1) HS LOGARIT:y =logax (x>0;a>0,a 1) TX: D=R Tp giỏ tr: (0; + ) Bin thiờn: a > 1: Hm s luụn ng bin 0 0, a 1) cú nghim nht x = logab b > 0, vụ nghim b Phng trỡnh logax = b (a > 0, a 1) luụn cú nghim x = ab vi mi b a a u v u v f ( x) f ( x) g ( x) logaf(x) = logag(x) a b u log a b v.log a b ,K:0 av u < v a>1 : au > av u > v a au av (a 1).(u v) Tp nghim Bpt k a>1 0< a < b0 R R ax> b b > x > logab x < logab b0 ax< b b > x < logab x > logab a >1: logaf(x) >logag(x) f(x) >g(x) >0 0a < x < ab < x ab Gia s Thnh c www.daythem.edu.vn O HM (e ) ' e x (e u ) ' u ' e u x (a u ) ' u '.a u ln a (a x ) ' a x ln a (ln x ) ' x (log a x ) ' (ln u ) ' x.ln a u' u (log a u ) ' u' u.ln a TH Hm s m y=ax ; TX : D=R Bng bin thiờn a>1 x y 0 < a 1 < a 0) CH í: t n ch nhm lm n gin bi toỏn nu hiu c thỡ khụng cn t n LOI 1: 1) 34x8 4.32x5 27 2) 22x6 2x7 17 sin x 4) 2.16 15.4 5) 7) 22x 3.(2x ) 32 1 8) ( ) x 3.( ) x 12 3 x x 10)25x-6.5x+1 + 53 =0 18) x x 2 x x 7.3 x x x 2 9.2 15) 20) 42 x 23x x2 16 23) x x 9) 9cos x 10 sin x x 6) 3.2sin x 11)32+x + 32-x = 12) 14) 52 x x 17) x x 3) x 3x 36.3x x 4.32 x2 9.2x x 2 13) 3x 9.3 x 10 x 2 x 16) x x 22 x x (H D 03) 2 19) 8x 3.4 x 3.2 x1 21) 22 x6 x7 17 22) ( ) x 65 x 12 24) 2sin x 4.2cos 25) 81sin x 81cos 2 x 2 x 30 LOI 2: Chia cho a m nh nht xong ri t t K : t > 1)3.4x-2.6x = 9x 2) 3.16x 2.81x 5.36x 3) 25 x 10 x 2 x 4) 5.4x 2.25x 7.10x 2 2 2 5) 252 x x x x 34.152 x x 6) 2.14x 3.49x x 7) 6.92 x x 13.62 x x 6.42 x x 8) 8x+4.12x18x2.27x=0 (A 06) 9) 32 x4 45.6 x 9.22 x2 LOI 3: (A*B=1) t t = ax m a > , K: t >0 1) (2 3)x (2 3)x x 4) 15 15 x 62 x x 2) 14 5) 24 3) 8) x 24 x x 52 x x 10 x 10 6) 7 x 7) ( 1) x ( 1) x 2 0.B 07 10) x1 (2 x 3x1 ) x1 x LOI 4: T N PH NHNG VN CềN x 1) x 2( x 2).3 x x 4) 3.16x2 (3x 10).4 x2 x 2) x.2 x 23 x x 3) x ( x 2).3 x 2( x 4) 5) 25x 2(3 x).5x 2x LOI 5: DNG HNG NG THC 1) x x x x 10 2) 31 x 31 x x x 3) 53 x 9.5x 27 53 x 5x 64 4) 23 x x x x 5) 8x1 8.(0,5)3 x 3.2 x3 125 24.(0,5) x 2 VD: Cõu 1) t t =(2x+2-x) , bỡnh phng v => 4x + 4-x = t2 DNG 3: CM PT cú nghim nht (s dng tớnh ng bin, nghch bin) 1) 2x 3x 5x 2) 3x 4x 5x 3)2x= 3-x 4)3x= 5- 2x 5)4x = x+2 6)6x + 8x = 10 x Gia s Thnh c www.daythem.edu.vn x 1) x 3x 2) x x 3) x x 25x 4) x 3x 5x 38 5) 8x 18x 2.27x 6) 3x 5x x 7) 5x x 8) x x 11x VD: Gii phng trỡnh: 3x 4x 5x (1) Cỏch 1: Ta thy x=2 l nghim ca phng trỡnh(1) Ta CM x=2 l nghim nht: (1) Vi x>2 ta cú: x>2 khụng l nghim Vi x 0,a 1); DNG 1: A V CNG C S : 1) log3 x log9 x log27 x 11 2)log3(5x + 3) = log3(7x + 5) 4) x log5 125 5x 25 5) log x 3x 7x 7) log x log x log 2x 9) log 0,5 (5 x 10) log 0,5 ( x x 8) 11) log6 x log3 x log36 x logaX = logaY 3) log5 x 11x 43 6) log9 x log9 x log9 2x 8) log3 x log x log3 x log x 10) ln( x 1) ln( x 3) ln( x 7) log ( x x 1) 12) log3 x log9 x log81 x 14)log2x + log2(x + 1) = 15)log4(x + 3) log2(2x 7) + 16) 13) log x log25 x log5 x log125 x Gia s Thnh c www.daythem.edu.vn 17) log2 x 16 log2 x 11 18) log ( x 2) log x log ( x 2) 19)log4x8 log2x2 +log9243 = 21) log x2 16 log x 64 20) log ( x x 18) 2log5 ( x 4) 1 22) log( x x 5) l og x log 5x 23)log0,2x log5(x 2) < log0,23 24) log log log (1 3log x) 25) log3 x 1 x 26) log5 x log5 x log5 x 29) log2x x 5x 27) log9 x log3 x 28)log2x.log4x.log8x.log16x = 30) log ( x 1) log ( x 3) log ( x 7) 31)log0,8(x2 +x +1)< log0,8(2x +5) 32) log x log x log8 x 13 33) log x 35) log (log2 x ) x 34) log2 (2x + 2) + log (9x - 1) = 37) log (3 x 5) log ( x 1) 36) log x x 5x 789 x3 38) lg( x x 3) lg x 3x x2 39)log 5-x(x -2x+65)=2 40) log x x log5 x 41) log x log 42)og2x1(2x2+x1)+logx+1(2x1)2=4 (H -A_08) S: x=2; x=5/4 44) log 4x 15.2 x 27 log 3x x2 43) log x 2log x log 0,4 (H_ D-07) S: x=log23 4.2 45) log3 (4 x 3) log x (H -A_07) S: 3/4 x 46) log3 x log x x x (H_B-08) S: 4< x < 3, x > x4 48) log 49) log5 4x 144 4log5 log5 x2 (H_ B-06) S: < x < 50) log x log x 47) log 0,7 log x 3x (H_D-08) S: 2;1 x 2; 53) log3 (log0,5 x) 52) log3 ( x x 12) 51) log2 ( x x 12) log2 (4x 6) 54) log2 [log (log5 x)] 3x x 55) log2 x 12 log2 x 12x 56) log2 (9x1 7) log2 (3x1 1) DNG 2: T N LOI 1: 1)2.log22 x - 14log4 x + 3= (TN-10) 2)log22(x - 1)2 + log2(x 1)3 = 3) log3 x log3 3x 4)log22x +log24x >0 5) log 2x log 2x3 8) 2log32 x 5log3 9x 6)ln3x ln2x = 4lnx -4; 7) log32 x 5log3 x 9) log3x 10 log 2x 10 6log x 10 11) log 22 x log x (H_D_08) S: x=1, x=3 13) 3logx 16 4log16 x 2log2 x 12) 10) 2log5 x log x 125 lg x lg x 14) log x log5x 2 15) log x log x Gia s Thnh c www.daythem.edu.vn 16) log2 x1 log2 x log 18) 13 log x 11 log x 12 21) log ( x2 x 2) 2log 17) log x log x 5x 2, 25 log 2x 19) log x log 4x x2 2x (*) 23) log x log2 ( x 1) 2 26) 4log9 x 6.2log9 x 2log3 27 20) log2 x 3log6 x log6 x 22) log22 ( x x 2) log1 ( x x 2) 24) log log x 125 x 25) log2 log3x x 27) 4log3 x 5.2log3 x 2log3 28) 2log2 ( x1) x log2 x 48 LOI 2: T XONG VN CềN X BI : BI 2: DNG 3: TNH N IU (CM PT cú nghim nht) 1) log2 x x 2) log x x 3) x log3 x 4) x log x 5) log x x 2 DNG 4: M HO DNG 5: BIN LUN THEO m Gia s Thnh c www.daythem.edu.vn Bi 1) Tỡm a phng trỡnh cú nghim nht : b lg ax a log3 x 4ax log 2x 2a lg x c log ( x mx m 1) log x0 d lg( x ax) lg(8x 3a 3) Bi 2) Cho ph-ơng trình: lg4 x 2m lg3 x mm lg2 x m m 1lg x m a) Giải ph-ơng trình với m = -1 b) Xác định m để ph-ơng trình có bốn nghiệm phân biệt Bi 3) Tỡm m phng trỡnh cú nghim tho : lg x m lg x m x loga ( x 2) loga x Bi 4) Gii v bin lun: Bi 5) Tỡm m tho: a log5 ( x 1) log5 (mx x m) ; x Bi 6) Tỡm m PT cú nghim thuc (0;1): b log2 (7 x 7) log2 (mx 4x m) ; x log2 x log x m Bi 7) Tỡm m PT cú nghim tho 4< x1< x2< 6: (m 3) log ( x 4) (2m 1) log1 ( x 4) m 2 Bi 8) H-A-2002 log32 x a) Gii PT m=2 log32 x 2m b) Tỡm m PT cú nghim trờn [1 ; 3 ] DNG 6: H PHNG TRèNH LOGARIT ( x y )x ( x y )y x y 25 log x log2 y lg x lg y 1) 2) 3) 24 4) 2 x 5y log x log y log2 x log2 y x y 29 log( x y ) log x y 11 log x log3 y log3 5) 6) 7) xy5 log x log y log 15 log( x y ) log( x y ) log log2 x log y log 8) log7 x y log2 ( x y ) 11) log ( x ) 1 y log2 ( x y ) 12) log4 x log2 y x y 15) log3 ( x y ) 2 log x y log xy 17) 3x xy y 81 x log8 y y log8 x 9) log4 x log4 y log2 log4 x log4 log2 y 10) log4 log2 x log2 log4 x x.2 y 1152 13) log ( x y ) log y log y x 14) x x y 20 log y x log y 16) (H_A 04) S: (3;4), (2;4) x y 25 (A 09) (2;2), (2;2) x y (B 05) (1;1), (2;2) 3log x log y 18) 10