Các phương pháp giải bài tập giải tích 12 nâng cao: Phần 2

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Nối tiếp nội dung phần 1 tài liệu Giải bài tập giải tích 12 nâng cao, phần 2 giới thiệu tới người đọc kiến thức cần nhớ và phương pháp giải các bài tập hàm số lũy thừa, hàm số mũ và hàm số logarit; nguyên hàm - Tích phân và ứng dụng; số phức. Mời các bạn cùng tham khảo.

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Nội dung Text: Các phương pháp giải bài tập giải tích 12 nâng cao: Phần 2

b) Tinh chat: Vdri a > 0, b > 0; m, n nguyen dUcfng va h a i so' p, q t u y y. ^jirmtg II« * 'Vab = '^.'l:a">a"m>n ^4^ 25 + 0 b"" o m < 0 1 * V d i a < b va n la so i\i n h i e n le t h i a" < b". / 1 3 s 5 ^ a) 81 -0,75 * V d i a, b l a cdc so duong, n e Z* t h i a" = b" o a = b. 125 l32j .5. 2. C a n b $ c n v a l u y thiifa v«Ji so m u hi^u t i 1 = 3-'' + ' .5-8 = i - 3 = - « « a) Dinh nghia: W6\ nguyen duotng, can bSc n cua so thUc a, k i h i ^ u l2 33 + 1 .,x3 27 27 27 la so thirc b sao cho b " = a. 5 • N/EI = b b" = a (n nguyen difong). 1 2 1 1 ? 1 ) 0,001 3 -(-2) ^643 -8 3 + ( 9 " ) ^ = (10"'')"3 - ( - 2 ) - 2 . ( 2 ' ' ) 3 -(2=) 3 +1^ Ghi chu: 111 • Neu n le t h i m o i so' thuc a chi c6 mot can bac n. = 10 - (-2rl2^ + 1 = 10 - 4 - — + 1 = 7 - — = 16 16 16 • Neu n chSn t h i m o i so' thUc duone a c6 h a i can bac n la >/a va - N/^ .
-ojr) 2 3 Cdc/i 2.- T a c6 7 + 5V2 = 1^ + 3.1^ 72 + 3.1.( N/2 )^ + ( >^ f = (1 + ^/3 - f 1^ -5 7 - 5V2 = 1^ - 3.1.72 + 3.1(72)^ - (72)^ = (1 - 73)^ c) + — 12, Dod6 7v + 572-+77-5v^ = 7(1 + 73)' +7(1-73)' -=9 + 8 - 5 = 12 3 = l + >,/3 + l - 73 = 2 8, ('Sdi 8, trang 78 SGK) Giai d) (0,5r* - ( 6 2 5 p ^ - + 19(-3)-' = -(5^) I 4. -27 7^-7b ^a+il^ (7^)^-(7b> (7^)^+7^' a) ^if^-ilh 7^ +7b 7a-7b 7^ + 7b = (2-')-^ - 5 - = 2* - 5 - = 16 - 5 - 1 = 10 12; 27 27 27 (7a" + 7b)(7i: - 7b) 7^(7"a + 7b) ,r 4/r 4r 4/r 5. (Bdi 5, trang 76 SGK) Giai 3u2 a^b^ _ __— — — —_ + Vb - Va = ^b a) - = ab (vdi a > 0, b > 0) b) 7a - 7b 7a + 7b 3 / 7 ^ a^b a-b a + b ^ (^f -(7b)' _ (^f + Cjhf 3 _5,3 7^-7b 7^ + 7b 7a + 7b 7^ + 7b b) (vdi a > 0) a + 7b) 4 2 1 ^ (7i - 7b)(7^ + 7 ^ + 7b^) (7^ + 7 b ) ( 7 ^ - 7 i b + 7b^) a^ ^ a^ a^ + a 3 i i 7^-7b" ^ a3(l-a') a 3 ( 1 - a " ) ^ (1 - a X l + a) ( l - a ) ( l + a) c) = 7^^ + 7Sb + 7 b ^ - 7 ^ + ^/Sb-7b^= 2^'^ ^^^^-7dbl:(7^-7b)^ aMl-a) a 3 ( a + l) ^'^ ,7a + 7b ) = 1 + a - ( 1 - a) = 2a (7^)'+(7b)' - T i b : ( 7 ^ - 27ib + 7b^) 6. (Bai 6, trang 76 SGK) Giai a) So sdiih N/2 va ^ (7ir + 7 b ) ( 7 ^ - 7 ^ + 7b^) T a c6: (>/2)' = 2 ' = 8; ( ^ ) ' = 3 ' = 9 - T i b :(7i^-27ib + 7b^) 7 i + 7b Do 9 > 8 nen (^f > {Sf suy ra ^ > V2 . = ( 7 ^ - 27ab + 7b^): (7i^ - 27ib + 7b^) = 1 b) >/3 + ^ va ^ a-1 7i+7i„l ,, ( 7 i - i ) ( 7 i +1) 7a(7I + 1 ) 4 r i • ^/3 + ^ > l + ^ = l +3 =4 a) —3 f.—]= .a" +1 = =—p= . == .Va +1 • ^ < ^ =4 J^J 7a+1 7 i ( 7 i + l) v'i + l (7i - i).7i.7i Vay 73 + ^ . >^ 1 = 7a - 1 + 1 = 7a 7i c) • W + A/15 < ^ + Vl6 = 2 + 4 = 6 9. (Bdi 9, trang 78 SGK) Giai • N/10 + ^ > V9 + ^ = 3 +3 =6 Tif tinh chat cua iQy thifa vdl so mu nguyen dUcfng, ta c6 Vay W + Vl5 < VlO + ^ . 7. (Bai 7, trang 76 SGK) Giai (Ta.Tb)" = (7a)".(7b)" = ab (vdi a > 0, b > 0, n nguyen duong). Cdch 1: Theo dinh nghia cSn h&c n cua mot 86' ta suy ra: Tab = Ta.Tb . Dat X = ^7 + 5^ + ^7-5sf2 = a+ b 10. (Bdi 10, trang 78 SGK) Giai Suy r a = a^ + b^ + 3ab(a + b) trong d6 a^ + b^ = 7 + 5 N/2 + 7 - =14 a) Cdch 1: ab = ^49 - 50 = - 1 74 + 273 - 74 - 273 = 7(73+1)' -7(73-1)' TO d6: x^ = 1 4 - 3 . 1 x = 73+ i | - 7 3 - i | = 73+ 1-73+ 1 = 2 o x^ + 3x - 14 = 0 0 ) o x = 2 Vay ^ 7 + 5>/2+^7-5>/2 = 2 . Dat X = 74 + 273 - 74-273 > 0
Ta c6 = 4 + 2>/3 - 2^4 + 2V3 .V4 - 2 7 3 + 4- 2>/3 b) T a CO 3^°° = (3^)^°° = 27^°° 5400 ^ (52)200 ^ 25^°° = 8- 2 V l 6 - 12 8- 2. ^/4 = 4 Suy ra X = 2 (do X > 0) Vay 3«"° > 5 Vay V4T2V3 - 4A-2S 2 1 c) = {2'')' =2' b) Cdc/i i ; ,2 Dat X = ^9 +VSO + ^9-^/80 =a +b 72.21" = 22.21" = 22'i" =21" =2 = a^ + b % 3ab(a + b) T r o n g do a ' + b^ = 9 + N/80 + 9 - 780 = 18 Vay = 72.2'" ab = ^9 + 7 8 0 . ^ 9 - 7 8 0 = 781 - 80 = 1 343 10 Tii do = 18 + 3.1.x 256 10 hay x^ - 3x - 18 = 0 >4^". (x - 3)(x^ + 3x + 6 ) = 0 o X - 3 = 0 (do x^ + 3x + 6 > 0) c=> x =1 3 Vay 780 + ^9 - 780 = 3 §2. L O Y THCTA V 6 I S O M U T H U C Cdc/i 2.- Do ^9 + 7 8 0 . ^ / 9 - 7 8 0 - 1 N e n neu ^ x/SO + 'iJ9~j80 = 3 t h i ^/gTTsO va vttOla hai nghieni I ^01 D U ] \ cAi\ cua phi/ong t r i n h - 3 X + 1 = 0. 1. K h a i n i ^ m 3 + 75. 3-75 Cho a l a so' thifc diTOng va a l a so' v6 t i Xi = 2 2 Xet day so' hufu t i r i , r2, T„, ... ma limrn = a 3 + 75 K h i do day so thuc a"' , a'', a'° c6 gidi h a n xac d i n h ( k h o n g phu (1) thuoc vao d a y (rn) da chon). T a goi gidi h a n do l a luy t h i i a ciia so' mu ot K i hay hieU a" ^ 3-75 ^9-780 = (2) a"= lima^" Ta chufng m i n h d i n g thiJc (1), t h a t vay Nhd: '3 + 7 5 ' ' 3^+3.3^75 + 3.3(75)'+(75)'^ - 7 2 + 3275 - L u y thCra v d i so' mu 0 va so' mu nguyen a m k h i co so' a 0. 8 8 - Luy thCfa vdi so' mu k h o n g nguyen t h i co so' a > 0 2. C o n g t h i t c l a i k e p = 9 + 4N/5 = 9 + 780 C = A ( l + r)^' T c r d o ^ = ^9+780 t r o n g do A l a so' t i e n gijfi r la l a i suat m o i k i Tifong tir = ^/g^Wlo . 2 N l a so k i gufi 11. (Bdi 11, trang 78 SGK) Gidi C so' t i e n t h u difoc ca vo'n l l n l a i 6 1 6 _5_ 1 BAlTAP 12. (Bdi 12 trang 81 SGK) Giai a) (73)"6 =(32)" 6 ^3 12 Chon B = V3 * = ( 3 =3 '2 (Bdi 13 trang 81 SGK) Gidi Chon (C) i 3 Vay(73)"« = 3 | 3 - > . 4 | i (Bdi 14 trang 81 SGK) Gidi Bieu k i e n 0 < a
15. (Bdi 15 trang 81 SGK) Gidi a' ' _ 1 d) (x" + y " ) ' - 4" xy = , / x ^ + 2x''y" + y^" - 4x''y'' • (0,5-'^)"' = 0 , 5 ^ = O.S'' = 16 2'^ 3v'5 23^''5 = 2x"y" + y ' " = v'^x" - y " ) " " - ! x" - y" I 20. TBa?; 2 0 trang 82 SGK) Gidi 16. r S a J 7P frung 81 SGK) Gidi a) - (a" + a-") = 1 o a" + a"" - 2 = 0 2 ,V5-3 • 4-V5 = a a ' 2 + a " 2 - 2 a 2 a 2 = 0 a^ - a zz 0 = a Ml) .V2 = a^^(a ')''^ ' = a^.a-^^' = a^^-^^-' = a' = a • N e u a ^ 1 t h i (1) o - = o a = 0 2 2 17. (Bdi 17 trang 81 SGK) Gidi • Neu a = 1 t h i (1) o a la so thuc t u y y Sau 5 n a m ngifcfi fi'y t h u duac ca von l l n l a i l a : b) 3'"' < 27 3'"' < 3^ » | a | < 3 -3 < a < 3 C = 15(1 + 0 , 0 7 5 6 f * 21,59 ( t r i # u dong) 21. (Bdi 21 trang 82 SGK) Gidi M HJYfeM T A P a) + i/x =2 18. (Bdi 18 trang 81 SGK) Dat t = ^ > 0 (v6i x > 0) T a CO phirang t r i n h t^ + t = 2 a) ilx^Vx = (x')v(x^)* = x2.x'2 = x'^''2 = xi2 (x>0) o t ^ + t - 2 = 0 c : > t = l hoac t = - 2 C5> t = 1 (do t > 0) '1 f ^* "> r 2 • Vdti t = 1, Vx = 1 X = 1 'a] ra^ 3 ~ 3 ' 15 'a > 0^ Vay phiTOng t r i n h da cho c6 mot n g h i e m x = 1 aVb = 1 lb. lb. ,b; b > 0, b) V i ' - 3 V ^ + 2 = 0 Dat t = > 0 (vdi x > 0) 2 /2 2 (2^ 3 (2^ 9 (2^ 18 (2' 3*9'18 ' 2 ^ IS '2^ C) 3 T a c6 phiWng t r i n h t^ - 3t + 2 = 0 o t = 1 hoac t = 2 — — — — — — — 3 V3V3 v3^ • Vdfi t = 1, ilx = 1 c:> x = 1 11 1 1 1 1 1; 1 ^1^1 ^ 1 11 d) \|a^ja'•Ja a 16 = ( a 2 ) . a ^ a « . a 16 . a^6 = a2 * 4 * 8 *16 . a 16 • V d i t = 2, = 2 X = 16 IB u 4 1 Vay phufOng t r i n h da cho c6 h a i n g h i e m x = 1 va x = 16. = a re ~16 =: a^e = a 4 42. (Bdi 22 trang 82 SGK) Gidi 19. (5di 19 trang 82 SGK) Gidi a) x ^ < 3 o 0 < x ^ < V3 0 - V 3
2. T i n h c h a t : C h o so' dUOng a 1 va cac so dUcrng b, c ^. (Bdi 27 trang 90 SGK) 'Giai * N e u a > 1 t h i logab > log^c b > c • log.-,3 = 1 * N e u 0 < a < 1 t h i logab > logaC b < c • l o g s S l = loggS" = 41og33 = 4 He qua: Cho so' difong a # 1 cdc s o ' dUcfng b, c • log.a = 0 • N e u a > 1 t h i logab > 0 o b > 1 I . log3 i = log.33-' = -21og33 = - 2 • • N e u 0 < a < 1 t h i logab > 0 b < 1 • logab = logaC C=> b = C • logs = log3 33 = ^ l o g 3 3 = ^ 3. C a c q u i tAc t i n h l o g a r i t Cho so dircrng a 1 va cAc so di/ong b, c, t a c6: • log3-V 3V3 = log33"'.3-"2 = logs 3 ^ = - f log.sS = - f 2 2 • lOga(bc) = logab + logaC 28. (Bai 28 trang 90 SGK) Gidi • loga - = logab - lOgaC \c) , l o g i 125 = l o g , = -31og i =-3 • logab" = alogab r 5 6 5 VO. 3. D o t cd so c u a l o g a r i t V d i a, b l a h a i s o difcJng khac 1 va c la s o dMng, t a c6: 2 • l o g o . s i = logl o = ^ loff c logbC = , hay l o g a b . l o g h C = logaC log = 31og, 1=3 logab 4 4 He qua: V d i a, b la h a i so difong khac 1, t a c6: 1 • logj36 = log, = - 2 log, 1 =-2 • logab = — hay logab.logba = 1 6 6 [6 6 29. (Bdi 29 trang 90 SGK) Gidi • log c = — logaC (a 0; c > 0) • 3'°*="* = 18 ^" a • 3^'"''' = 3'°*^^''' = 2^ = 32 4. L o g a r i t t h a p p h a n f-\ L o g a r i t co s o 10 cua inQt s o dirong x difcfc goi la i o g a r i t t h a p p h a n cua x. {8^ 125 k i hieu Igx (hay l o g x ) log, 2 / Blog,2 log, 2 ' logn,5 2 ^^ * Logarit thap phan c6 day du cac t i n h chat cua logarit v d i ccf so lorn hem 1. 2 (V = 2'* = 32 ^ B A IT A P • . [32J .2) [2,. J V 23. (Bai 23 trang 89 SGK) Giai 30. (Bdi 30 trang 90 SGK) Gidi Chon d). a) logsx = 4 CO x = 5'' = 625 24. (Bai 24 trang 89 + 90 SGK) Gidi b) log2(5 - x ) = 3 c = . 5 - x = 2^c=.x = - 3 a) Sai b) D i i n g c) Sai d) Sai. c) log3(x + 2) = 3 X + 2 = 3'^ X = 25 25. (Bai 25 trang 90 SGK) Gidi (1' ' d) l o g , (0,5 + X) = - 1 o 0,5 + X = 0,5 + X = 6 X = 5,5 a) loga (xy) = logaX + logay 6 (Dieu k i e n a > 0, a 1; x > 0, y > 0) 31. (Bdi 31 trang 90 SGK) Gidi b) loga— - logaX - logay (Dieu k i ^ n a > 0, a 1; x > 0, y > 0) . log,25 = . 1,65 y log? c) logax" = alogaX ( D i l u k i e n a > 0, a 1, x > 0) d) a'"^'" = b (Dieu k i e n a > 0, a * 1, b > 0) . log,8 = . 1,29 log5 26. (Bai 26 trang 90 SGK) Gidi a) logaX < logay 0 < X < y . l o g , 0 , 7 5 . ^ . - 0 , 1 3 log9 Dieu k i e n a > 1 b) logaX < logay x > y > 0 • logo,. = 1 ^ « -0,42 log0,75 Dieu k i e n 0 < a < 1
b) l o g l 2 - log5 = l o g — = log2,4 32. (Bdi 32 trang 93 SGK) Gidi do log2.4 < log? . a) log8l2 - logglS + logg20 = logs~~ = logglG = log^,, 2' = ^ Vay l o g l 2 - log5 < log? 15 o c) 31og2 + log3 = log2'^ + logs = log2^3 = log24 b) ^ log736 - log7l4 - 31og7 ^ 21og5 = logS^ = log25 do log24 < l o g 2 5 = log? (6^)2 - log7l4 - l o g 7 ( 2 1 ^ ) ' = log72.3 - log72.7 - log73.7 v a y 31og2 + logs < 21og5 = log72 + log73 - (log72 + log,?) - (Iog73 + log77) d) 1 + 21og3 = loglO + logS^ = l o g i c s ^ = log90 ^ l0g72 + l0g73 - log72 - 1 - l0g73 - 1 = - 2 Do log90 > log27 Cdch khdc Vay 1 + 21og3 > log27 35. (Bdi 35 trang 92 SGK) Gidi ^ log736 - log7l4 - 31og7 ^ 1 a) logax = log«a%' x/c = logaa-'.b'. = l0g76 - I0g7l4 - l0g721 = l 0 g 7 _ ^ = l0g77^' = - 2 14.21 = logaa' + log„b^ + logaC^ = 3 + 21og„b + - l o g . c = 3 + 2.3 + - ( - 2 ) = 8 36 c) l " g r , 3 6 - l o g , 1 2 ^ ' " g ^ i 2 ^ logs 3 ^ 1 b) logaX = l0g« ^ logaa^ + l o g , - log^ log, 9 log,3' 21og5 3 2 Cdchkhic l " g r . 3 6 - l o g , 1 2 ^ l o g ^ =iog,3(d6icas6)= log.,3 = ilog33 = 4 + ^ log.b - SlogaC = 4 + i .3 - 3 ( - 2 ) = 11 logs 9 logs 9 " 2 3 3 36. (Bdi 36 trang 93 SGK) Gidi a) logsx = 4Iog3a + 71og3b logsx = logaa" + log3b^ => logsx = logsa^.b'' => x = a*.b^ = 6'"^'^^ + — - 2'"^^'^ = 25 + 5 - 27 = 3 2 b) logox = 21og5a - 31og5b Cdc/i khdc => logsx = logoa'^ - loggb^ logsX = l o g 5 ^ => X = ^ = 6'"^-'' + 10'"«''^ - 2'°^-^^' = 25 + 5 - 27 = 3 37. (Bdi 37 trang 93 SGK) Gidi 33. (Bdi 33 trang 92 SGK) Gidi a) log^gSO = log .5.10 = 2[log3 5 + log3lO] a) log34 > log33 = 1 B i e t log3l5 = log33.5 = loggS + log35 = 1 + logaS l o g 4 i = log43-' = -log43 < 0 Suy r a log35 = a - 1 3 Vay log ,| 50 = 2[a - 1 + p] = 2a + 2p - 2 Vay log34 > l o g 4 1 3 b) Iog4l250 = log,, 5^2 = laogzS" + log22| = ^ [41og25 + 1| = 2a + | b) 3'°«»''' v& 7'°^«"'-'' Do l o g s l , ! > 0 n e n 3 ' * ' ' > 3" (do 3 > 1) 38. (Bdi 38 trang 93 SGK) Gidi hay 3'°^'-'' > 1 1 1 ' r • ' T' va log60,99 < 0 n e n 7'"'''''''^ < 7" (do 7 > 1) a) l o g - + - l o g 4 + 41og72 = log2^^ + I o g ( 2 ' ) 2 + log 2^ = log2^l2.2^ = l o g l hay 7'*"-^''< 1 Vay 3'°^>i''' > y'nKsOSf l b ) l o g ^ + ^ l o g S 6 + | l o g ^ = log(2^3-') + log(6')5 + log 34. (Bdi 34 trang 92 SGK) Gidi 9 2 v2 J 3 3 a) log2 + log3 = log2.3 = log6 V2 o-2^ = Iog(2'.S"') + log(2.3) + logCSl 2 " 2 ) = log(2l3-^.2.3.3^ 2"^) do log6 > log5 1 V a y log2 + log3 > log5 = log(3^.2.22 ) = log(18.72 )
c) log72 - 2\og~ + logN/108 §4. SO e VA LOGARIT lU NHIEN 3 = l o g ( 2 l 3 ' ) - log + log(2^3^)2 = l o g ( 2 l 3 ' ) - log(3^2-'«) + log(2.32) v2 . IVOI D I J I ^ G cAl^ NH€i 3 = log ^ ' - f = log(2^°. 3'^ ) = log2^" + log3'^ = 201og2 - | log3 I, So e: e = l i m 2,71828 ... 2. Cong thiJc l a i kep l i e n tuc S = A.e''^ d) l o g - - log 0,375 + 21ogV0,5625 8 A: so' vo'n ban dau r: l a i sua't m 6 i n a m = log 2-' - log(0,5^3) + log(0,5^32) ^ i o g l ^ | 5 _ 3 _ ^ i ^ g ^ = log — N : so n a m S: so t i e n t h u dugc ca vo'n I a n l a i 39. (Bdi 39 trang 93 SGK) Gidi 3. L o g a r i t tu" n h i e n Dieu k i e n x > 0, x 1 L o g a r i t cof so e cua mot so diTcfng a difoc goi la logarit t i i nhien (hay logarit a) logx27 = 3 c o x ^ = 2 7 c o x = 3 n e - p e ) cua so a. K i h i e u Ina * L o g a r i t t i f n h i e n c6 day du t i n h chat cua l o g a r i t co so lorn hon 1. b) l o g : , ! = - 1 o X"' = ! o X " ' = 7"' o X= 7 7 7 J BAITAP ^ 1 - i 42. (Bdi 42 trang 97 SGK) Gidi c) log, V5 = - 4 o X-' = >/5 X = (VS) ^ « X = 5 » Sai tuf ln(2e) = Ine + Ine 40. r S d i 40 trang 93 SGK) Gidi V i ln(2e) = l n 2 + Ine ^ Ine + Ine M31 = 2^' - 1 43. (Bdi 43 trang 97 SGK) Gidi So cdc chuf so cua M 3 1 k h i v i e t t r o n g he t h a p p h a n b^ng so cac chuf so cua 2 " • ln500 = l n 5 ^ 2 ' = ln5'^ + l n 2 ' = 31n5 + 21n2 = 3b + 2a 16 nen so cac chuf so cua M 3 1 la [31.1og2] + 1 = [9,31 + 1 = 10 • I n — = l n 2 ^ 5 - ' = l n 2 ' + l n 5 - ' = 41n2 - 21n5 = 4a - 2b Tuang tiS, so cac chC so cua M 1 2 7 = 2^^'' - 1 k h i v i e t t r o n g he t h a p p h a n la: [127.1og2] + 1 = 38 + 1 = 39 •ln6,25 = I n — = l n = 2[ln5 - l n 2 ] = 2(b - a) 100 .2 So cdc chuf so cua Mi398269 k h i v i e t t r o n g he t h a p p h a n la 1,. 1 1 2 , 98 , 99 [1398269.1og2] + 1 = 420921 '"^ o + '^^^T + ... + I n — + l n 2 3 99 100 41. (Bai 41 trang 93 SGK) Gidi = I n l - l n 2 + l n 2 - l n 3 + ... + ln98 - ln99 + ln99 - InlOO So t i e n ca von iSn l a i sau n qui \k = I n l - InlOO 0 - l n ( 2 . 5 ) ' = - 2 ( l n 2 + ln5) = - 2 ( a + b) S = 15(1 + 0,0165)" = 15.1,0165" ( t r i e u dong) 44. (Bdi 44 trang 97 SGK) Gidi Suy r a logS = l o g l 5 + nlogl,0165 — ln(3 + 2 V2 ) - 41n( ^/2 + 1) - ^ l n ( 72 - 1) _ logS-logl5 ^ logl,0165 ^= ^ ln( 72 + If - 21n( 72 + 1)^ - ^ l n ( 72 - 1)^ 16 16 De CO dugc 20 t r i e u dong t h i p h a i sau mot t h d i gian la 25 |-^ln(72 +l)'^-^ln(72 -1)^ n = . 17,58 (qui) J-D 16 logl,0165 = - | J [ l n ( 7 2 + l ) ^ + l n ( 7 2 - 1)2] Vay sau 4 n a m , 6 t h a n g (4 n a m , 2 qui) ngifdi gufi se c6 i t nha't 20 t r i e u tiT so 16 von 15 t r i e u dong ban dau) ;-^ln[(72 +1)1(72 - l ) ^ j = - ^ l n ( l 2 ) = 0 (vi sau qui h a i , ngifdi gijfi m d i n h a n dUc/c l a i ) •lo 16
4 6 . (Bai 45 trang 97 SGK) Gidi Ti le t a n g trufdng m 6 i gicr ciia lohi v i khuS'n PC D a o h a m c i i a h a m so m u Tii cong thufc S = A.e''' * H a m s6 y = a" C O (3ao h a m l a y ' = a''.lna 300 = lOO.e^'' * H a m so y = e" c6 dao h a m l a y ' = e" ^ ^ ^ l n _ 3 0 0 - l n l 0 0 ^ ]n3 ^ * H a m so y = a"'"* c6 dao h a m la y ' = u'(x).a""'\lna u(x) * H a m so' y = e c6 d a o h a m la y ' = u'(x).e' 5 5 D a o h a m c u a h a m so' l o g a r i t T i 1§ t a n g t r u d n g ciia loai v i k h u a n n^y la 21,97% m 5 i gid. Sau 10 gid, tii 100 con v i k h u a n se c6: * Ham s o ' y = logaX (vdi x > 0) c6 d a o h a m y ' = xlna TiJf 100 con, de c6 200 con t h i thori gian c^n t h i e t l a * H a m so' y = I n x ( v d i x > 0) c6 dao h a m y ' = — X 200 = lOO.e^-^'^^' u'(x) , In200-lnl00 ln2 o-,r: o •>,n u - . sd' y = logaU(x) (vdi u(x) > 0) c6 dao h a m y ' = uix).lna =>t= = « 3,15 gift = 3 gid 9 p h u t * Ham 0,2197 0,2197 u'(x) 46. (Bdi 46 trang 97 SGK) Gidi * H a m so' y = Inu(x) ( v d i u(x) > 0) c6 dao h a m y ' = u(x) T i le p h a n hiiy h a n g n a m ciia Pu^'^ Ta c6: S = lO.e''^^^*'" Dac b i $ t : y = I n i x | (vdfi m o i x 0) t h i y ' = — Suy ra r = ^"^ " ^"^^ « -2,84543.10^^ ~ -0,000028 y = I n I u(x) I (vdfi m o i u(x) 0) t h i y ' = — — 24360 u(x) Vay svf p h a n h u y ciia Pu^^** AKac t i n h theo c6ng thuTC 4. b i e n t h i e n v a do t h i c u a h a m so m u , h a m so l o g a r i t g _ ^ g-0,000028t a) Ham so mu y = a' T r o n g do S va A t i n h b k n g gam, t t i n h b&ng n a m . • T $ p x&c d i n h : M V a i 100 g a m Pu^''^, t h d i gian cAn t h i e t de p h a n h u y con 1 gam la • D o n g b i e n t r e n K k h i a > 1; nghich bien t r e n R k h i 0 < a < 1 1 = 10 e"'^'^^^'^^*'' • Do t h i ] n l - j n l O ^ 82235 (nam) - D i qua d i e m (0; 1) -0,000028 - Nhm p h i a t r e n true h o a n h Vay sau k h o a n g 82235 n a m t h i 10 g a m chat Pu^'*'' p h a n hiiy con 1 gam. - N h a n true h o a n h l a t i e m can ngang \' §5. HAM SO MU VA HAM SO LOGARIT - a I I \ O I DUI\ CAN MlIC? 0 1 1. K h a i n i $ m v e h a m so m u v a h a m so l o g a r i t Dang dS thi * Cho a > 0 va a 1 b) Ham so y = logaX • T a p xdc d i n h la (0; +oo) - H a m so' y = a" duoc goi la h a m so' mu co so' a • Dong bien tr§n (0; +oo) k h i a > 1, nghich bien t r e n (0; +oo) k h i 0 < a < 1 - H a m so y = logaX dJoc goi l a h a m so l o g a r i t co so a • Do t h i * H a m so' y = a'' va y = logaX l i e n tuc t a i m o i d i e m m a no duac xac d i n h . - D i qua d i e m ( 1 ; 0) • Vxo e M, l i m a" = a'^ . - N k m b e n p h a i true t u n g X->Xo - N h a n true t u n g la t i ^ m can diJng • Vxo e (0; +oo); l i m logaX = logaXo * M p t so g i d i h a n : l n ( l + x) , ,. e^-l , 1 • lim = 1; h m = 1 x-*o X X 0 Dang dS thi
^ BAITAP 51. (Bdi 51 trang 112 SGK) Gidi a) - Ham so y = (x/2 )" xac dinh tren R y 47. (Bdi 47 ti-ang 11 SGK) Gidi a) Khi nhiet do ciia nUdc la 100"C thi luc do P = 760 - Ham so dong bien tren R (ca so J2 > 1) 2- -2258.624 Do thi hkm so: Do do ta c6: 760 = a. 10'"°^"'' - Di qua cdc diem (0; 1); (1; ^); (2; 2) •—TV -2258.624 2258.624 - N^m phia tren true hoanh . 1 2 hay 760 = a. 10 ^'^ ; a = 760.10 - Nhan true hoanh lam tiem can ngang -2258.624 Vgly a « 863188841,4 40*273 b) Ham so' y = xac dinh tren b) Tinh ap sua't htfi nifdc: p = a. 10**"'^ - 863188841,4.10 [3) 2 y 22M.624 - Ham so nghich bien tren E (co so — < 1) = 863188841,4.10 SI'S" ^ 52,5 mniH. 3 48. (Bai 48 trang 112 SGK) Gidi . e^-e^-^ ,. e ^ ( l - e ^ - ) Do thi ham so '2-^1 a) lim = lim I ' 3j V 9j - Di qua cdc diem (0; 1); -1 0 1 ^ x = hm = -3e.-hm — = -3e -r 1 2 - Nam phia tren true hoanh _ g6x3x x ^ go2 x x-u 3x - Nhan true hoanh la tiem can ngang = 2 - 5 = -3 trang 112 SGK) Loai am Gidi 52. (Bdi 52STT thanh I Do Idtn (L) b) lim = lim \. x-->0 +0 49. (Bai 49 trang 112 SGK) V X Gidi 1 NgiTSng nghe 1 0 dB a) y = (x - De'" ^ y' = e'^ + (x - l).2e''' = e'^ (1 + 2x - 2) - (2x - l)e'^ 2 Nhac em diu 4000 36 dB 3 Nhac manh phat ra tCf loa 6,8.10** 88 dB b) y = x ' Ve'" + 1 4 Tieng may bay phan liic 2,3.10'^ 124 dB y' = ix'y4^^ +xl(N/e^^+l)' = 2xN/e^' D b) y'y == V( V x U^ ^l ^.Inx^ )Mnx'+Vi^^.(lnx2)'= y3j 3 .Inx^ + 7 ? T l . ^ H^m s6' y = .>/2 + V3 nghich bien tr§n M (vi co so x.lnx' 2NA? +1 2Vx^ +1 ^ V2 + Vs < 1) = +
c) y = x . l n - 1 +X §6. HAM SOLUYTHLfA -1 y' = x ' . l n 1 + x + x. I n -1 + x . = I n -1 + x + X. (i + x7 1 + x 1. K h a i n i ^ m h a m so l u y thxia InCx'' +1) - H a m so luy thCfa la h a m so' dang y = x" (trong do a la h a n g so') d)y = - H a m s6' y = x°, v6i a k h o n g nguyen xdc d i n h t r e n (0; +oo) 2x 2 2. D a o h a m c u a h a m so l u y thvta , [ln(x^ + l ) ] ' . x - l n ( x ^ + l ) . x ' ^T^-^-^^^^ y = = - H a m so y = x" (vdi a e K ) c6 dao h ^ m y' = a x " " ' - H a m so' y = u°(x) (v6i a G K) C6 dao h a m y' = a u ° ' kx).u'(x) 55. (Bdi 55 trang 113 SGK) Gidi a) H ^ m so y = logaX nghich b i e n t r e n (0; + x ) 3. V a i n e t ve su" b i e n t h i e n do t h i h a m so l u y thtifa e * H ^ m so l u y thiTa y = x" (vdi a 0) x^c d i n h t r e n (0; + x ) 2 * H a m so dong b i e n t r e n (0; +t3o) n§'u a > 0 v^ nghieh bien t r e n (0; +x) v i cc( so' — < 1 e neu a < 0 b) H ^ m so y = log«x dong b i e n t r e n (0; +•«) • Do t h i h a m so qua ( 1 ; 1) vdfi m o i a 1 V3 + x/2 Mot s6 dang dd thi VI cd so a = > 1 3(73 - N/2) 3 56. (Bdi 56 trang 113 SGK) Gidi 1 a) - H ^ m s6' y = log ,^ x xdc d i n h t r e n (0; +oo) - H a m so d 6 n g b i e n t r e n (0; + « ) O 0 1 Do t h i h ^ m so - D i qua d i e m ( l ; 0 ) , ( ^ ^ ; l ) - Nkm bgn p h d i true t u n g - N h a n true t u n g la difdng t i | m ean diJng M B4I TAP b) H ^ m so y = l o g j X xac d i n h t r e n (0; + x ) m. (Bdi 57 trang 117 SGK) Gidi 3 Gia sijf (Ci) va (Cj) I a n lucft l a do t h i cac h ^ m so y = x" va y = x" (a, p c6 the - H ^ m s6' nghieh b i e n t r e n (0; + 1 ta eo bat d^ng thijfc - N l i m ben phSi true t u n g x" > x° o p > a - N h $ n true t u n g la t i 0 m can dufng Do do j3 = - - va a = - 2 2 Vay (C2) la do t h i h a m so y = x 2 (Ci) 1^ do t h i h ^ m so y = x ^ ^8. (Bdi 58 trang 117 SGK) Gidi a) y = (2x + D ' y' = 7i.(2x + D " - ^(2x + D ' = 27i.(2x + 1)"" ' b) D a t u = In'Sx u' = 3.1n25x.an5x)' = S.ln^Sx. ^ ^^"'^"^ 5x
60. (Bai 60, trang 117 SGK) Gidi Ta CO y' = ( ^ ) ' = i .u^ ' . u ' = - 5u» a) Goi ( C i ) va (C2) M n lugt la do t h i cdc h ^ m so y = ax^ va y = 1 31n'5x 31n'5x Vay y' = M(xo; yo) l a d i e m bat k i . D i e m doi xufng ciia M qua true t u n g l a M'(-xo; yo), 5^(ln^'5x)^ • X 5xVln"5x 5x^n'5x ta c6: /JN c) D a t u = M e (Ci) o yo = a"" o yo = M' e (C). 1-x^ _ 3 x ^ ( l - x ' ^ ) + 3x^(l + x^) _ 6x^ Dieu nay chufng to ( C i ) va (C^) doi xiifng nhau qua true tung. b) Goi (C,'j) va ( C 4 ) I a n luot la do t h i cae h a m so y = log^x va y = logj x . Do do y ' = (u^'")' = - u ^ u ' = — ^ .u' = Goi M(xo; yo) la d i e m bat k i . * D i e m doi xutng cua M qua true hoanh la 3 3 ^ 3u M"(xo; -yo). Ta CO M e (C3) yo = l o g a X o c=> y„ = - log, x„ o M" e (C4). 6x^ l ^ _ _ 2 x ^ 3 [ l + x^ a D i e u n ^ y chufng to (C;J), ( C 4 ) do'i xufng nhau qua true hoanh. l-x' 61. (Bai 61, trang 118 SGK). Gidi a ' x ^ ^a^ ^ a) H a m so y = logonx xdc d i n h t r e n (0; +oc). y' d)y = (v(5i a > 0, b > 0) lb. • H a m so' nghich b i e n t r e n (0; + « ) . A" Do t h i h a m so: a X 3 + x - D i qua di§m ( 1 ; 0); (0,5; 1). h) Ax b; b IbJ vx; - N a m ben p h a i true tung. b / \ a ^xV b X a a [ X - N h a n true t u n g la t i e m can dufng. .b. b b. X lb; Vx V X J x lb . a) Can cuf vao do t h i , t a c6: logo,5X > 0 .3- b r 0 < X < 1 a_b fa-b •(ling v6i phan do t h i d phia t r e n true hoanh). X X bj .X, X ; I b) - 3 < logo.sx < - l c:>2
b) IOg2X + log2(x - 1) = 1 (2) §7. PHLfdNG TRINH M O V A LOGARIT ....... ... vtfi dieu k i ^ n fx>0 X > 1 X - 1> 0 (2) log2x(x - 1) = 1 » x(x - 1) = 2' x^ - X - 2 = 0 o x = - 1 h o a c X = 2 (cau a) 0 t h i phu'cfng t r i n h a" = m x = l o g e m . a) Theo g i a t h i e t k h i d = 0 t h i F 53 ( k H z ) + Phitcfng trinh logarit cd ban: log^x = m k h i d - 12 t h i F 160 (kHz) • logaX = m a"' = X ( d i l u k i e n xac d i n h x > 0). Ka"=53 (1) Ta c6: 2. M p t so phifcfng p h a p g i a i K a ' ' = 160 (2) Ta giai mot so dang philcfng t r i r i h mu (hoac phUcmg t r i n h logarit) b k n g m p t so Tif (1) s u y r a k = 53 phUcfng phap. * DUa cac luy thiia (hoac cac logarit) trong phUcmg t r i n h ve luy thiia (hoac logarit i (2) suy r a 5 3 . a ' - = 160 cung cd so. 12 160 ,2 a = o a" = 3,019
b) 0 , 1 2 5 . 4 ' " - ^ = (4>/2 )" 1 A2X 1 94x 6 o4 Dat t = f-T > 0 , t a c6: t^ + t = 2t3 + t - 2 = 0 « > ( t - l)(t^ + t + 2) = 0 22 t = 1 (do t^ + t + 2 > 0) « 2^" = 2^ « - X = 9 C5 X = 6 ^3^ 2 t = 1, = 1 o X= 0 V a y p h u a n g t r i n h c6 n g h i e m l a x = 6 V a y n g h i e m ciia phUcfng t r i n h l a x = 0. 6T. (Bdi 67, trang 124 SGK) Gidi 69. (Bdi 69, trang 124 SGK) Gidi a) logzx + log4X = l o g i N/3 ( D i ^ u k i e n x > 0). 2 a) log^x^ - 201og7x + 1 = 0 . Dieu k i e n x > 0. 1 « log,x + 1 ^ = 1 ^ (ddi s a n g CO so' 2) o 91og-x - 201ogx2 + 1 = 0 91og^x - lOlogx + 1 = 0 D a t t = logx, t a c6 phi/ofng t r i n h 1 3 i 9 t ' - lot + 1 = 0 » t = 1 hoac t = i o logzx + - l o g . x = -log2 7 3 o - l o g ^ x = -log^S^ 9 2 i 2 1 . 1 • t = 1, logx = 1 X = 10 O l0g2X = log2(32) 3 O 10g2X = l o g j S 3 o x = 3 3 1 1 i • t = - , logx = - < = > x = : 1 0 9 < = > x = 7 l O x = (thoa dieu kien) 73 N g h i e m p h i i o n g t r i n h 1^ x = 10 x = ^|lO . logjX logg4x PhUcfng t r i n h c6 n g h i e m = • b) . Dieu k i e n x > 0. log,2x logie8x b) logjgX.loggX.loggX = 8 . D i e u k i e n x > 0. logj^x Iog2X ^ 3 « i ^ . l o g 3 X . ^ =8 o l ^ . l o g 3 X . ^ =8 log2 2 x logjSx log3V3 \og,2 1 2 2 21og,x ^ 4(loga4 + l o g a X ) 21og, x _ 4 (2 + l o g , X) o Iog'3X = 2 ^ o log3X = 2 o x = 3 ^ 0 X 3 9 log2 2 + l o g 2 X Sdog^S + log^x) l + l o g , x ~ 3(3 + log,,x) V a y phi/ong t r i n h c6 n g h i e m x = 9. 2t 4 (2 + t) 68. (Bai 68, trang 124 SGK) Gidi D a t t = log2X, t a c6 phifong t r i n h : 18 1 +t 3 ( 3 + t) o 6t(3 + t ) = 4(1 + t)(2 + t ) 18t + et"* = 4(t^ + 3 t + 2) a) 3""' + 18.3"" = 29 o 3.3" + ^ = 29 o 6t^ + 18t - 4t^ - 12t - 8 = 0 3t^ - 29t + 18 = 0 o t = 9 hoac t = - 16 3 • t = 9, 3" = 9 0 Phirong t r i n h c6 h a i n g h i e m x = 2 va x = log3 2 - 1 1 I log9x27 - log3,3 + log9243 = 0. Dieu k i e n X * - b) 2 7 " + 12" = 2.8"
X = 2 X = 2 ^ logaQx logaSx loggS X = - -1 x = -log32-l (thda dieu ki#n) logaQ + loggX l O g g S + lOggX 2 Phuong trinh c6 hai nghiem x = 2 va x = -(log32 + 1). 3 1 X > 0 d) x ^ 5 = 5 ' . Dieu kien 2 + log3X l + loggX 2 x^^l Dat t = logax, ta co: _ i - - : ^ + | = 0 X = o X® = 5'°'^'-' o log,x^ = log,(5""'«^5-') » 6(1 + t) - 2(2 + t) + 5(2 + t ) ( l + t) = 0 ^6= log^5 + l o g , 5 - ' » 6 = log^5-51og,5 6 + 6t - 4 - 2t + 5t^ + 15t + 10 = 0 Dat t = logx5 « 5t' + 19t + 12 = 0 Ta CO p h u c f n g t r i n h t^ - 5t - 6 = 0 o t = -1 hoac t = 6 4 t = -3 hoac t = - - • t = 1, logx5 = - l < = > x = 5 < = > x = —. o 5 . t = - 3 , logsx = -3 « X = 3 - ^ = ^ • • t = 6, logx5 = 6x^ = 5x = Vs. Vay p h U c m g t r i n h c6 hai nghif m x = - x = N/S .t = - i , l o g 3 X = - | c . x = 3 ^ « x = ^ - o x = 5 '81 (Bdi 71, trang 125 SGK) Gidi a) 2^ = 3 - X Phiitfng t r i n h c6 hai nghiem x = — va x = - DS t h a y X = 1 l a n g h i | m phifotng t r i n h (vi 2^ = 3 - 1). Ta c h u f n g m i n h p h i f c f n g t r i n h k h o n g c6 n g h i e m n^o khac. 7 0 . (Bdi 70, trang 125 SGK) Gidi - Ham so 7 = 2" dong bien tren K . a) 3^' = e - Ham so' y = 3 - X nghich bien tren R . ^ loggS^" = log34^" » 4''log33 = 3Mog34 * Vdi X > 1 ta c6 2" > 2^ = 2 Vdrix>ltac6 3 - x < 3 - l = 2 o 4^ = 3Mog34 o = log.,4 o X = log4(log3 4) Do d6 2" > 3 - X tren k h o a n g (1; +»). 3, Khong CO gid t r i nao cua x thoa phucfng trinh da cho. Phifong trinh c6 nghiem 1^ x = log4 (log34) * Vdi X < 1 t a CO 2" < 2' = 2. it x < l t a c 6 3 - x > 3 - l = 2. b) 3^ '"^'^ = 81x , dieu kien x > 0 Do do 2" < 3 - X tren khoang (-oo; 1). « Jl_ = 3^x « - = 3\ « x 2 = ^ « x = i (dox>0) Khong CO gia t r i nao ciia x thoa phiTcfng t r i n h da cho. Vay phuorng t r i n h da cho c6 nghiem duy nhat la x = 1. b) log2X = 3 - X. D i l u k i ^ n xdc dinh x > 0. Phirong trinh c6 mot nghi$m x = - • D l t h a y x = 2 l a n g h i e m d u y n h a t c u a phiTcfng t r i n h (vi log22 = 3 - 2l = l ) . Ta c h u f n g m i n h p h U c f n g t r i n h k h o n g c6 nghiem nao khac. c) 3\' ' = 36. Dieu ki§n x * - 1 . - Ham so' y = log2X dong bien tren (0; +
§8. HE PH J O N G TRiNH MU VA LOGARIT b) (II) - y^' = 2 l o g j C x + y ) - log3(x - y ) = 1 V d i dieu ki§n x + y > O v ^ x - y > 0 . BUNG CAI\ 'log2(x + y) + l o g 2 ( x - y ) = 1 (4) Gidi he phucmg t r i n h mu vk logarit ta dung cdc phufcfng phdp the, cong dai log2(x + y ) - l o g 3 ( x - y ) = 1 log2(x + y ) - l o g 3 ( x - y ) = l (5) so, dat phu... TriJf tCfng v e h a i phUcfng t r i n h (4) (5). ^ BAITAP log2(x - y ) + log3(x - y ) = 0 72. (Bdi 72'trang 127 SGK) Gidi o log2(x - y ) + ^"^2^^ = 0 log2(x - y ) 1 + X + y = 20 (1) log2 3 = 0 a) (I) logjSj log4X + log4y = l + log49 (2) Dieu kien (x > 0; y > 0). o log2(x - y ) = 0 (do 1 + -1— ^0) \og,3 Xet phiJong t r i n h (2) log4X + log4y = 1 + log49 X-y=lci.x=l + y log4X.y = log44.9 oxy = 36 Tii phiTcmg t r i n h - = 2 fx + y = 20 Do do (I) i [xy = 36 (vdi X > 0, y > 0) o (1 + y)^ - y2 = 2 « y^ + 2y + 1 , y2 = 2 o y = i X, y la nghiem phuong t r i n h - 20X + 36 = 0 • Tir do: X = - (thoa d i l u kien). X = 18 hoac X = 2 Vay tap nghi|m h$ phifdng trinh (I) S = 1(2; 18); (18; 2)1 Vay tap nghiem cua he phiTcfng trinh Ik S = '3 V] X +y = 1 (3) b) (II) 4 - 2 X + 4 - 2 y ^ Q 5 (4) 2' 2 74. (Bai 74, trang 127 SGK) Gidi Tii phifcfng t r i n h (3): y = 1 - x a) log2(3 - x) + l o g z d - x) = 3 (1) The vao phifang t r i n h (4): 4-^" + 4-2'^-"' = 0,5 o 4-2" + 4'\4'^'' = 0,5 Vdi dieu kien x < 1. Dat t = 4'" > 0 (1) » log2(3 - x ) ( l - x) = log223 Ta c6: - + — t = - c=> (3 - x ) ( l - x ) = 8 < = > x ^ - 4 x + 3 - 8 = 0 t 16 2 o 16 + t^ = 8t t^ - 8t + 16 = 0 o (t - 4)^ = 0 o t = 4 (thoa dieu kien) P x2-4x-5 = 0 « ''""V, [x = 5 (loai) t = 4, 4^" = 4 o 2x = 1 o x = - Vay tap nghiem phUdng trinh S = | - 1 | . b) log2(9 - 2") = 10'°^='-'" (2) 1 1 H Vdi dieu kien x < 3. x = — => y = — 2 - ^ 2 ^ ( 2 ) o logaO - 2") = 3 - X o log2(9 - 2") = log22'-'' Vay tap nghiem cua h f phi/cfng t r i n h (II): S = a 1 • 0. log^(x + y) = 2 (2) B o 9 t - t ^ = 8 o t ^ - 9 t + 8 = 0 o t = l hoac t = 8 Tii phircfng t r i n h (2) log^(x + y) = 2 • • t = 1, 2 " = 1 o X = 0 • • t = 8, 2" = 8 « 2 " = 2^ X= 3 (loai) x + y = (%/5)^x + y = 5y = 5 - x (3)i H Phuong t r i n h c6 nghiem x = 0. The (3) vao (1): 3"\2^-'' = 1152 m 7iogx _ giogx+i ^ 3 5iogx-i _ i3_7iogx-i (3) « 3-\2^2-'' = 1152 6-" = i ^ l ^ o 6 " = 6 ' « - X = 2 « X -2 B Vdi dieu kien x > 0. 2^ B (3) « . + 13.7'°^-^ = 3.5'"^"^ + 5'°^*' Tii (3): y = 7 B ^ ylogx ^ 2^ ^logx _ 3 gi^g, ^ g Vay tap nghiem h§ phuorng t r i n h S = l ( - 2 ; 7)). • 7 5
7.28 c) 5 Vlog2(-x) = log.N/^ (3) 20 7'°*" = 28 7 5 l5y 5.20 V6i d i l u k i e n x < 0. '7^ logx r7f logx = 2 logx = loglO^ D a t t = log2(-x); T a c6 \og^^ = log^lxj = log2(-x) (do x < 0) o t > 0 .5> l5J (2) o 5>/t = t CO X = 100 (thoa dieu kien). '25t = t ' Vay phuong trinh c6 nghiem x = 100. ft > 0 t > 0 d) 6^ + 6''^^ = 2" + 2"^! + 2"*' lt(t-25) = 0 o t = 0 hoac t = 25 o 2''.3'' + 6.2^3'' = 2^+2.2" + 4.2" t = 0 hoac t = 25 o 7.2''.3'' = 7 . 2 " « 3" = 1 (do 7.2" > 0) • t = 0; log2(-x) = 0 o - x = l o x = - l (thda d i l u k i e n ) . o X = 0 • t = 25; log2(-x) = 25 o - X = 2^^ o X = -2^^ (th6a dieu k i $ n ) , V$y t a p n g h i $ m cua phuctog t r i n h la S = (-2^^ - 1 } . Nghiem phiTOng trinh lA x = 0. 75. (Bdi 75, trang 127 SGK) Gidi Vdri d i l u k i e n x > 0. a) log3(3" - l)log3(3''*i - 3) = 12 (1) V6i dieu ki|n x > 0. (4) ^ 3"*«'".32 +3'°^*".3'2 = o 3'°*«".(32 +3'^) = 4^ (1) o log3(3" - l)[log33(3'' - 1)] = 12 ^/3 log4X = l o g a - ^ « log3(3" - 1)[1 + log3(3'' - 1)] = 12 V3 l3 3 v3 D a t t = logsO" - 1). Ta CO phucmg trinh t(l + t) = 12 o X= 4 (thda digu ki§n). o t ^ + t - 1 2 = 0 o t = 3 hoac t = - 4 10g2 V|ly t a p n g h i S m cua phifcfng t r i n h la S = ( 4 . t = 3, l o g 3 ( 3 " - l ) = 3 76. (Bai 76, trang 127 SGK) Gidi
r 2I (Bdi 77, trang 127 SGK' Gidi V a y t $ p n g h i f m cua phiTcrng t r i n h S = - ^ l o g — • • a) 2""'" + 4.2^°^'" = 6 Inx+l b) 4 - 6'"" - 2.3" = 0 (2) « • 2""'" + 4.2'"''"'" = 6 0 2""'" + 4.2.2"''"'" - 6 = 0 V d i d i l u k i | n x > 0. Dat t = 2'"''" do 0 < sin^x < 1 n e n 1 < t < 2, t a c6 phiTcJng t r i n h . (2) o 4'"\ - 2'"''.3'"'' - 2.3^'"^3^ = 0 X ^Inx gin's 321nx = 0 t + 8. i - 6 = 0 (v 1. Nen - < t < 4, t a c6 phurcmg t r i n h (3) S^log^ X - (logj 8 + l o g j x) + 1 = 0 o S^logjX - 3 - logaX + 1 = 0 o 37log^X - l o g g X - 2 = 0 t = l 64.t^ - 28.t - 2 = 0 o 32.t^ - 14t - 1 = 0 o 2 D a t t = yJ\og^x, t > 0 t = - ~ (loai) T a c6 phuong t r i n h : 3 t - t ^ - 2 = 0 o t ^ - 3 t + 2 = 0 16 t = l o (th6a dieu k i $ n ) . • t = - 4'^*'^" = - t = 2 2 ' 2 • t = 1, ^ l o g j X = 1 log2X = 1 X = 2 271
Ngn: 3.U + 2 . V = 2 , 7 5 ^ rsu = 1 , 2 5 fu = 0,25 X -1 • Ng'u X < - 1 thi > — u - V = -0,75 |u - V = - 0 , 7 5 |v = 1 ( t h o a d i e u k i # n ) l3> x +4 < - 1 + 4 = 3 • u= 0,25 2 " = 0 , 2 5 2 " = i « 2" = 2"^ « X = - 2 4 / I A" dodd > X + 4 .v=lo3y=ly = 0 V a y n g h i e m h | phUOng t r i n h : (x; y) = ( - 2 ; 0 ) . Dieu n a y chilng to t r e n k h o a n g (- - 1 thi = 3 v&x + 4 > - 1 + 4 = 3 3 + log.,y = log2 5 ( l + 31ogsx) (2) 13J l3j V d i d i e u ki§n x > 0 , y > 0 . do d6 < X + 4 {3) log2 2^ + log2 y = log2 5 + 31og2 x [logj 8y = log^ 5 + Slogg x D i l u n^y chiJng to t r e n khodng ( - 1 ; + « ) phuong t r i n h (1) k h o n g c6 nghiem 5X^ i n Vay X = - 1 1^ n g h i e m duy nha't cua phUong t r i n h da cho. logjxy = log6l0 xy = 10 x . — - = 10 f TTY f TtV logaSy = log2 5.x^ 8y = 5x^ 5x= b) sin- + cos- =1 (2) y =• \ y 5J x' =16 X = 2 (do x > 0) D l t h a y x = 2 l a m o t n g h i e m cua phifcfng t r i n h v i s i n ' - + cos' - = 1 . X = 2 5x^ « 5x^ y =• y = y = 5 T a chtJng m i n h phUcfng t r i n h ( 2 ) k h o n g c6 n g h i e m n^o k h ^ c , t h a t vky do 0 < s i n - < l ; 0 < c o s ^ < l , nen: V a y n g h i ? m ciia h ^ phi/Ong t r i n h 1^ (x; y) = ( 2 ; 5 ) . 5 5 X f . Ttl . n sm— < sm— < 5j § 9 . BAT P H l / d N G TRINH M O V A L O G A R I T • Ng'u X > 2 thi - X < ^^ ( I l>fOI D U I V G CAM NH6 cos— < cos— .1 5, I 5j G i i i b a t p h u o n g t r i n h mO b a t phiTong t r i n h l o g a r i t , c i n nhdf c d c h ^ r a so . 71 71 dod6 sm— + cos— < 2 . Phifong t r i n h ( 2 ) khdng c6 nghiem tr§n ( 2 ; +xi y = a" y = log^x d o n g b i e n k h i a > 1 n g h i c h b i e n k h i 0 < a < 1. 5 , 5J 1 BAITAP . 7t . 71 80. (Bdi 80, trang 129 SGK) Gidi sm— a) 2^-^^ > 1 « 2 3 - ^ " > 2 ° • N&'u X < 2 thi . 5J sm— 3 - 6x > 0 (do t i n h d o n g b i e n c u a h a m so mQ ccf so 2) cos— 1 cos— ^ ^ < 2 - 71: do d6 sm- cos- 3— 5j Tap n g h i ^ i i i ciia b a t p h u o n g t r i n h 1^ S = (~oo; 1). PhUOng t r i n h (2) k h o n g >c6 2 n g h i $ m t r e n (-=»; 2). b) 16" > 0,125 V a y X = 2 1^ n g h i e m duy n h a t ciia phUOng t r i n h da cho. o 16" > - o 2 ' " > 2'^ 79. (Bdi 79, trang 127 SGK) 8 3.2" + 2.3^ = 2,75 o 4x > - 3 (do t i n h d o n g b i e n c u a h ^ m m u ccf so 2) a) ( I ) 3 2" - 3* = -0,75 o X > — 4 Dat u = 2^ V = 3'' (u > 0; V > 0). Q T a c6 h? phuctng t r i n h Tap n g h i e m c u a bd't phUOng t r i n h 1^: S = ( - - ; + » ) 4
I. (Bdi 82, trang 130 SGK) Gidi 81. (Bai 81, trang 129 SGK) Gidi a) logogX + logggX - 2 < 0 . Dilu kien xac dinh x > 0. a) loggOx - 1) < 1 (1). Dieu kien xac dinh 3x - 1 > 0 Dat t = logo.sx logsOx - 1) < 1 rr, 9 ft > -2 logsCSx - 1) < logsS 3x - 1 < 5 (do tinh dong bien ciia h^m so logarit CO so 5). Ta c6: t'' + t - 2 < 0 » -2 < t < 1 (1) '3x - 1 > 0 3 o - 1 o 2"0 X2° 0 (do tinh dong bien ciia h^m so mu co t >(2)» logo,5(x' - 5x + 6) > - 1 so 2). X logo.50,5'' o x^ - 5x + 6 < 2 (do tinh nghich bieii Tap • t 2bat ' ' 0SGK) Gidi x' - 5x + 6 > 0 Jx^ - 5x + 6 > 0 a) logo,i(x^ + X - 2) > logo,i(x + 3) (1). Dieu kien xdc dinh , x^ + X - 2 > 0 ( 3 ) o x' - 5x + 6 < 2 jx' - 5x + 4 < 0 [x + 3 > 0 X < 2 hoac X > 3 1 -3 l - 2 x < 1 (do tinh dong bien ciia ham so logarit la co so 3). -yfE < X < >/5 1 0 j Tap nghi$m bat phiiong trinh da cho la S = ( -V5; - 2) u (1; >/5). X 0 0 (2) (4)» X < 0 hoSc X ^ 3- l-2x < 1 x^ - 6x + 5 > 0 x Dieu kien xdc dinh 2 - x > 0 Tap nghi^m bat phUdng trinh da cho la S = 1 .3' 2) r
l o g i ( x ' - 6x + 5) + 21og3(2 - X ) > 0 -1 + 2 ( 2 ' . 2 - ) » l o g i (x^ - 6x + 5) > -log3{2 - x ) ' » l o g , ( x ' - 6x + 5) > l o g , (2 - x ) ' 3 3 3 1+"-(2" + 2 - ' ) 2" + 2 " + 2 « - 6x + 5 < (2 - x)^ (do t i n h n g h i c h b i e n h ^ m so l o g a r i t cor so - ) . o 2 ' ' + 1 - 2.2' (2- -1) 2' - 1 1-2" (do X < 0 n e n 2" < 1). 2x - 1 > 0 2'" + 1 + 2.2" ~ V(2'' +1)' 2" + 1 1 + 2" 86. (Bai 86, trang 130 SGK) Gidi x ' - 6x + 5 > 0 x < 1 hoSc x > 5 a) A = 92'0 x < 2 o - 0, c - b > 0). 4 Nen V 4 => 21ogaa = log,(c - b) + loga(c + b) => log„(c - b) + log3(c + b) = 2 1 + . ll + T ( 2 X -(2" + 2 " ) ' 4 + — - — = 2 => loge+ba + logc-bB = 21ogc-ba.log,.^ba loge-ba log„ba