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Gio vin: Nguyn Thnh Long Email: [email protected]: 01694 013 498

1

(DNG CHO N THI TN C H 2011)

Gi tng: www.Mathvn.com

Bm sn. 11.04.2011

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2

CC PHNG PHP GIIPHNG TRNH - BT PHNG TRNH - H M - LGARIT

CHNG I:PHNG PHP GII PHNG TRNH - BT PHNG TRNH - H MCH I: PHNG TRNH M

BI TON 1: S DNG PHNG PHP BIN I TNG NG

I. Phng php:

Ta s dng php bin i tng ng sau:

Dng 1: Phng trnh f x g xa a

TH 1: Khi a l mt hng s tha mn 0 1a th f x g xa a f x g x

TH 2: Khi a l mt hm cax th

1

0 1f x g xa

aa a

f x g x

hoc

0

1 0

a

a f x g x

Dng 2: Phng trnh:

0 1, 0

logf x

a

a ba b

f x b

c bit:

Khi 0, 0b b th kt lun ngay phng trnh v nghimKhi 1b ta vit 0 0 0f xb a a a f x

Khi 1b m b c th biu din thnh f xc cb a a a f x c

Ch :Trc khi bin i tng ng th f x v g x phi c ngha

II. Bi tp p dng:

Loi 1: C s l mt hng s

Bi 1: Gii ccphng trnh sau

a. 1 11

12 .4 . 16

8x x x

x

b.

2 3 11

33

x x

c. 1 22 2 36x x

Gii:a. PT 1 2 2 3 3 42 2 6 4 4 2x x x x x x x

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3

b.

2

23 1

( 3 1) 1 21 3 3 3 ( 3 1) 13

x x

x xx x

2 13 2 02

xx x

x

c. 1 22 8.2 2

2 2 36 2.2 36 364 4

x x xx x x

x x 49.2 36.4 2 16 2 4x Bi 2: Gii ccphng trnh

a. 2 32

0,125.48

x

x

b. 2 1

718 0, 25 2

xx

x

c. 2 2 3 32 .5 2 .5x x x x

Gii:

Pt

122 32

3

1 2

. 28 2

x

x

5 5 53 2(2 3) 3 4 6 4 92 2 2 52 .2 2 2 2 2 2 4 9 6

2

x

x xx x x

x x x

b. iu kin 1x

PT2 1 73 2

21 2

12 1

2 2 3 7 2 7 9 2 0 21 2

7

x x

x

xx x

x xx x

c. Pt 2 3

2.5 2.5x x

2 310 10 2 3 1x x x x x

Bi 2: Gii phng trnh: 3log1

2 22

x

x x x

Gii:Phng trnh cho tng ng:

33loglog

3

2 0 22 0

111 log ln 0ln 01222

222 0

xx

x xx

x xxx

xxx

3

2 2 2

log 0 1 121 1 3

ln 0 12 2 2

2 22

x x x

x x x

xx x x

x xx

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4

Bi 3: Gii ccphng trnh:

a. 3 1

1 310 3 10 3x x

x x

b.

21 1

3 22 2 4x

x x

Gii:a. iu kin:

1

3

x

x

V1

10 310 3

.

PT 3 1

2 21 3 3 110 3 10 3 9 1 51 3

x x

x xx x

x x xx x

Vy nghim ca phng trnh cho l 5x

b. iu kin:0

1

x

x

PT

2 32 22

2 131 12 12 2 4 2 .2 4

x

x xxx xx x

2 32

1 2 12 32

2 4 21 2 1

4 2 3 4 1 4 10 6 0 3 9

x

x x xx

x x x

x x x x x x x x

Vy phng trnh c nghim l 9x

Loi 2: Khi c s l mt hm ca x

Bi 1: Gii phng trnh sin 2 3cos2 22 2

x

x x x x

Gii:Phng trnh c bin i v dng:

2

2

2

1 2(*)2 0

1 0(1)2 1 sin 2 3 cos 0

sin 3 cos 2(2)

xx x

x xx x x x

x x

Gii (1) ta c 1,21 5

2x tho mn iu kin (*)

Gii (2):1 3

sin cos 1 sin 1 2 2 ,2 2 3 3 2 6

x x x x x k x k k Z

nghim tho mn iu kin (*) ta phi c:

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5

1 11 2 2 1 2 0,

6 2 6 2 6k k k k Z

khi ta nhn c 3 6

x

Vy phng trnh c 3 nghim phn bit 1,2 31 5

;2 6

x x

.

Bi 2: Gii phng trnh: 22 43 5 2 23 6 9

x xx xx x x

Gii:

Phng trnh c bin i v dng: 2

2 243 5 2 2 2( 4)3 3 3

x xx x x x

x x x

2 2 2

3 1 44

0 3 1 3 45

3 5 2 2 2 8 7 10 0

x xx

x xx

x x x x x x

Vy phng trnh c 2 nghim phn bit x = 4, x = 5.

Bi tp t gii c hng dn:

Bi 1: Gii cc phng trnh sau

a.2 1

1 24.9 3.2x

x

b. 1 2 4 37.3 5 3 5x x x x

c. 4 3

745 4 327 3

x x

x x

d.

31 13 1 1x x

x x

HD:

a.

2 33 3

1 22

x

x

b.1

1 1 33 5 1 15

x

x xx

c. 10x

BI TON 2: S DNG PHNG PHP LGARIT HO V A V CNG C S

I. Phng php:

chuyn n s khi s m lu tha ngi ta c th logarit theo cng 1 c s c 2 v ca phng trnh, ta c

cc dng:Dng 1: Phng trnh:

0 1, 0

logf x

a

a ba b

f x b

Dng 2: Phng trnh: (c s khc nhau v s m khc nhau) ( ) ( ) ( )log log ( ) ( ).logf x g x f x f xa a aa b a b f x g x b

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6

hoc ( ) ( )log log ( ).log ( ).f x g xb b ba b f x a g x

c bit: (c s khc nhau v nhng s m bng nhau)

Khi

0

( ) 1 0f x

f x f x a af x g x a b f x

b b

(v ( ) 0f xb )

Ch : Phng php p dng khi phng trnh c dng tch thng ca cc hm m

II. Bi tp p dng:

Bi 1: Gii ccphng trnh

a. (H KTQD 1998)1

5 .8 500.x

x x

b.2

2 323 .4 18

x

x x

c.2 4 22 .5 1x x d.

2 2 322

x x

Gii:a. Cch 1: Vit li phng trnh di dng:

1 1 333 2 385 .8 500 5 .2 5 .2 5 .2 1

x x xx x xx x

Ly logarit c s 2 v, ta c:

3 3

3 32 2 2 2 2

3log 5 .2 0 log 5 log 2 0 3 .log 5 log 2 0

x x

x xx xx

xx

22

31

3 log 5 0 1

log 5

x

xxx

Vy phng trnh c 2 nghim phn bit:2

13; log 5x x

Cch 2: PT

33( 1) 3 13 2 3 35 .2 5 .2 5 2 5 2

xx x

x x xx x x

331

311

5

3 0 315 5.2 1

log 25.2 12

xx

x x

xx

x x

x

b. Ta c2 2

2 3 2 32 2

3 33 .4 18 log 3 .4 log 18x x

x xx x

2 23 3 34 6 3( 2)

2 .log 2 2 log 2 4 .log 2 0x x

x xx x

2 3 23

2 02 2 3log 2 0 2

2 3log 2 0 ( )

xx x x x

x x VN

c. PT2 4 2

2 2log 2 log 5 0x x

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7

2 2 24 2 log 5 0 2 2 log 5 0x x x x

2 2

2 2

2 log 5 0 2 log 5

x x

x x

d. Ly logarit cs 2 hai v phng trnh ta c:2 2 2 2

2 2 2 23log 2 log 2 log 3 1 2 1 log 3 02

x xx x x x

Ta c , 2 21 1 log 3 log 3 0

suy ra phng trnh c nghim x = 1 2log 3.

Ch :i vi 1 phng trnh cn thit rt gn trc khi logarit ho.Bi 2: Gii ccphng trnh

a. 428 4.3x

xx b.1 1

2 12 24 3 3 2x

x x x

c. 9

14

)2cossin52(sin5,0

log

xxx

d. 1 2 3 15 5 5 3 3 3x x x x x x

Gii:a. iu kin 2x

PT 3

242

2 2

3 12 3 2 (4 ) log 3 4 . log 3 0

2 2

x

xx x x xx x

2 3

4 04

1log 3 0 2 log 2

2

xx

xx

b.

PT1 1 1

2 1 2 2 23 4

4 2 3 3 4 . 3 .2 3

x x xx x x

3 3

2 23

4 3 0 02

x x

x x

c. iu kin 2sin 5sin .cos 2 0 *x x x

PT 1 2 242log sin 5sin .cos 2 log 3x x x

22 2log sin 5sin .cos 2 log 3x x x tha mn (*)

2 cos 0

sin 5sin .cos 2 3 cos 5sin cos 0 5sin cos 0

22

1tan tan

5

x

x x x x x x x x

x kx k

x lx

d. PT

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8

5 5.5 25.5 3 27.3 3.3

531.5 31.3 1 0

3

x x x x x x

x

x xx

Vy nghim ca phng trnh cho l 0x

Bi 3: Gii ccphng trnha. lg 21000xx x b. 2 4log 32xx

c. 2

25 5log 5 1 log 77 x x d. 13 .8 36x

x x Gii:a. iu kin 0x

22lg .lg lg1000 lg lg 2 lg 3 0

lg 1 0 1 /10lg 1 lg 3 0

lg 3 0 1000

x x x x x

x xx x

x x

b. iu kin 0x PT 2 4log2 2 2 2 2 2log log 32 log 4 .log 5 log 1 . log 5 0x

x x x x x

2

2

2log 1

1log 5

32

xx

x x

c. iu kin 0x

225 5log 5 1 log 7 2

5 5 25 5 5 5

52 25 5 5 5

5

log 7 log log 5 1 .log 7 log 7.log

1log 11

log 5 log 1 0 log 2 log 3 0 5log 34 125

xx x x

x xx x x x

x x

Vy phng trnh cho c nghim1

5125

x

x

d. iu kin 1x

12 2 2 2 2

22 2 2

2 2 2 23

3log 3 .8 log 36 2 2log 3 .log 3 2 2log 3

1

.log 3 3 log 3 2 1 2 1 log 3

2.log 3 1 log 3 2 2log 3 0 1 log 2

x

x xx

xx

x x x x

xx x

x

Vy phng trnh c nghim l:3

2

1 log 2

x

x

Bi 4: Gii cc phng trnh sau :

a.2 1 18 .5

8x x b. 1

43 . 9

27x x

x

c. 12.32

xx d. 22 .5 10x x

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9

Gii:a. Ly logarit hai v vi c s 8, ta c

2 21 18 8

1 18 .5 log 8 .5 log

8 8x x x x

2 1 1 2

8 8 8 8log 8 log 5 log 8 1 log 5 1

x x

x x

2 8 81 1 log 5 0 1 1 1 log 5 0x x x x x

8 8

1 01 1 1 log 5 0

1 1 log 5 0

xx x

x

8 8 5

1 1

.log 5 log 5 1 1 log 8

x x

x x

Vy phng trnh c nghim: 51, 1 log 8x x

b.PT 2 2 3 2 2 33 .3 .3 4 3 4 2 2 log 4x x x x

x

3 3 3 3

3

42 log 4 2 2 log 4 log 9 log9

1 4 2log log

2 9 3

x x

x

c. Ly log hai v ca phng trnh theo c s 2

Ta c phng trnh2 2

2 2 2log 3 log 2 0 log 3 0x x

x x

22

0( log 3 ) 0

log 3

xx x

x

d. PT2 2

2 2 2 2 2 2log (2 .5 ) log (2.5) log 2 log 5 log 2 log 5x x x x

2 22 2 2 2

2

2

log 5 1 log 5 (log 5) 1 log 5 0

1

1 log 5

log 5

x x x x

x

x



a. 15 . 8 100xx x

HD: iu kin 0x 2( 1) 3 2( 1) 2( 1) 2 2

22

5

5 .2 5 .2 5 2

2log 5.( 2) 2

1 log 2( )

x x x x x x x x

xx x x

x loai

b.2 23 2 6 2 52 3 3 2x x x x x x

HD:

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10

2 ( 2)( 4)2

3

2 3 2 ( 2)( 4) log 3

2

log 2 4

x x xx x x

x

x


a.2

3 .2 1x x b.2 4 22. 2 3x x c.

2 5 6 35 2x x x d.1

3 .4 18x

x x

e. 228 36.3x

xx f. 7 55 7x x

g. 53 log5 25x x i. log 54 3.5 5 xx k. 9log 29. xx x s:a. 30; log 2 b. 32;log 2 2 c. 53;2 log 2 d. 32; log 2

e. 34; 2 log 2 f. 7 55

log (log 7) g. 5 h. 41

; 55

k. 9

BI TON 3: S DNG PHNG PHP T N PH - DNG 1

I. Phng php:

Phng php dng n ph dng 1 l vic s dng 1 n ph chuyn phng trnh ban u thnh 1 phngtrnh vi 1 n ph.Ta lu cc php t n ph thng gp sau:Dng 1: Phng trnh ( 1)1 1 0..... 0

k x x

k k a a

Khi t xt a iu kin t > 0, ta c: 11 1 0...... 0k k

k kt t t

M rng: Nu t ( ) ,f xt a iu kin hp 0t . Khi : 2 ( ) 2 3 ( ) 3 ( ), ,.....,f x f x kf x ka t a t a t

V ( )1

f xat

Dng 2: Phng trnh 1 2 3 0x xa a vi a.b 1

Khi t ,xt a iu kin t 0 suy ra1x

bt

ta c: 221 3 1 3 20 0t t tt

M rng: Vi a.b 1 th khi t ( ) ,f xt a iu kin hp 0t , suy ra ( )1f xbt

Dng 3: Phng trnh 2 21 2 3 0xx x

a ab b khi chia 2 v ca phng trnh cho 2 0xb ( hoc

2 , .xx

a a b ), ta c:2

1 2 3 0x x

a a

b b

t ,x

at

b

iu kin 0t , ta c: 21 2 3 0t t

M rng:

Vi phng trnh m c cha cc nhn t: 2 2, , .ff f

a b a b , ta thc hin theo cc bc sau:

- Chia 2 v phng trnh cho 2 0fb (hoc 2 , .ff

a a b )

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11

- tf

at

b

iu kin hp 0t

Dng 4: Lng gic ho.Ch : Ta s dng ngn t iu kin hp 0t cho trng hp t ( )f xt a v:

- Nu t xt a th 0t l iu kin ng.- Nu t

2 12xt th 0t ch l iu kin hp, bi thc cht iu kin cho t phi l 2t . iu kinny c bit quan trng cho lp cc bi ton c cha tham s.

II. Bi tp p dng:

Bi 1: Gii phng trnh

a.2 2

1cot sin4 2 3 0x x (1) b.

2 2sin cos4 2 2 2x x Gii:a. iu kin sin 0 ,x x k k Z (*)

V 22

11 cot

sinx

x nn phng trnh (1) c bit di dng:

22 cotcot4 2.2 3 0g x

x (2)

t2cot2 xt iu kin 1t v

22 cot 0cot 0 2 2 1xx Khi phng trnh (2) c dng:

22 cot 212 3 0 2 1 cot 03

cot 0 ,2

xt

t t xt

x x k k Z

tho mn (*)

Vy phng trnh c 1 h nghim ,2

x k k Z

b. PT 22 2sin 1 sin2 2 2 2x x

t 2sin2 0xt t ta c

2 3 22 2 2 2 2 2 0 2 2 2 0t t t t t t t

2

2 2 4 22

2 2 4 22

t

t

t loai

Vi1

2 221 2sin2 2 2 sin sin2 2 4 2

xt x x x k

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12

Vi22 2 4 2 sin2

2xt

(phng trnh v nghim)


a.

7 4 3 3 2 3 2 0

x x

b. (H B 2007) 2 1 2 1 2 2 0x x

c. 33 5 16 3 5 2x x

x

d. (HL 1998) sin sin

7 4 3 7 4 3 4x x

e. 5 24 5 24 10x x

Gii:

a. Nhn xt rng:

2

7 4 3 2 3 ; 2 3 2 3 1

Do nu t 2 3

x

t iu kin t 0 , th: 12 3x

t v 27 4 3

x

t

Khi phng trnh tng ng vi:

2 3 2 213

2 0 2 3 0 1 3 03 0( )

tt t t t t t

t t t vn

2 3 1 0

x

x

Vy phng trnh c nghimx = 0

b. t 2 1x

t ta c Pt:

12 2t

t 2 2 2 1 0t t 2 1 2 1t t 1 1x x

c. Chia 2 v ca phng trnh cho 2 0x , ta c:

3 5 3 5

16 82 2

x x

Nhn xt rng:3 5 3 5

12 2

t3 5

2

x

t

, iu kin t > 03 5 1

2

x

t

Khi pt (*) c dng:

2

3 5

2

3 58 16 0 4 4 log 4

2

x

t t t x

d. Nhn xt rng: 7 4 3. 7 4 3 7 4 3 7 4 3 1

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13

t sin

7 4 3x

t , iu kin t > 0 sin 1

7 4 3x

t

Khi pt (1) c dng:

sin2 1sin

2

sin sin2

2 3 2 37 4 3 2 32 31 4 4 1 0

2 3 7 4 3 2 3 2 3 2 3

xx

x x

tt t tt t

sin 1

sin

2 3 2 3 sin 1cos 0 ,

sin 1 22 3 2 3

x

x

xx x k k Z

x

e. Nhn xt rng: 5 24 5 24 1

t

5 24

x

t , iu kin t > 0

15 24

x

t

Khi pt (1) c dng:

1

25 24 5 24 5 24 5 245 241

10 10 1 05 24 5 24 5 24 5 24 5 24

x x

x x

tt t t

t t

1

1

x

x

Nhn xt:

-Nh vy trong v d trn bng vic nh gi:

27 4 3 2 3 ; 2 3 2 3 1

Ta la chn c n ph 2 3x

t cho phng trnh

- Vic la chn n ph thng qua nh gi m rng ca a.b 1 , l: . . 1a b

a b cc c

tc l vi cc phng

trnh c dng: . . 0x xA a B b C Khi ta thc hin php chia c 2 v ca phng trnh cho 0xc , nhn c:

. 0x x

a bA B C

c c

t thit lp n ph , 0x

at t

c

v suy ra1

xb

c t


a. (HTL 2000)2 22 1 2 22 9.2 2 0x x x x

b.2 2 21 1 12.4 6 9x x x

Gii:a. Chia c 2 v phng trnh cho 2 22 0x ta c:

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14

2 2 2 22 2 1 2 2 2 21 92 9.2 1 0 .2 .2 1 02 4

x x x x x x x x 2 22 22.2 9.2 4 0x x x x

t2

2x xt iu kin t 0 . Khi phng trnh tng ng vi:2

2

2 22

21

42 2 2 1

2 9 4 0 1 212 22

x x

x x

tx x x

t t xt x x

Vy phng trnh c 2 nghim 1 2x x .b. Bin i phng trnh v dng:

2 222 1 2 112.2 2.3 3x xx

Chia hai v ca phng trnh cho 22 1

2 0x

, ta c:

2 21 2 13 3

22 2

x x

t

2 1

32

x

t

, v

2 1 1

2 3 3 31 12 2 2

x

x t

Khi pt (*) c dng:

2 12 2

3 32 2

2 32 0 2 1 log 2 log 2 1

1 2

xtt t x x

t l

Ch :Trong v d trn, v bi ton khng c tham s nn ta s dng iu kin cho n ph ch l 0t v chng ta

thy vi1

2t v nghim. Do vy nu bi ton c cha tham s chng ta cn xc nh iu kin ng cho n

ph nh sau:

22 1

2 44

1 1 1 12 2

2 4 4 2x x

x x x t


a. (HYHN 2000)

33 1

1 122 6.2 1

22x x

xx

b. (HQGHN 1998) 3 1125 50 2x x x Gii:a. Vit li phng trnh c dng:

33

3

2 22 6 2 1

2 2

x x

x x

(1)

t33

3 33

2 2 2 22 2 2 3.2 2 6

2 2 2 2x x x x x

x x x xt t t

Khi phng trnh (1) c dng: 32

6 6 1 1 2 12

x

xt t t t

t 2 , 0xu u khi phng trnh (2) c dng:

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15

2 1 ( )1 2 0 2 2 2 122

xu loaiu

u u u u xu

Vy phng trnh c nghimx = 1b. Bin i phng trnh v dng:

125 50 2.8 1x x x

Chia hai v ca phng trnh (1) cho 8 0x , ta c:

3 2

125 50 5 52 2 0 2

8 8 2 2

x x x x

t5

2

x

t

, iu kin 0t

Khi pt (2) c dng:

3 2 22

1 52 0 1 2 2 0 1 0

2 2 0 2

xtt t t t t x

t t VN

Bi 5: Gii cc phng trnh

a.

2 11

1 13. 12

3 3

x x

b. 13 3 4 0x x c. 1 4 24 2 2 16x x x

Gii:a. Bin i phng trnh v dng:

2 1

1 112 0

3 3

x x

t1

3

x

t

, iu kin 0t

Khi pt (1) c dng:

23 1

12 0 3 14 3

xtt t x

t loai

b. iu kin: 0x

Bin i phng trnh v dng:3

3 4 03

x

x

t 3 xt , iu kin 1t

Khi pt (1) c dng:

2

14 3 0

3

t loait t

t loai

c. Bin i phng trnh v dng: 2 1 4 22 2 2 16x x x

22.2 6.2 8 0 1x x

t 2xt , iu kin 0t Khi pt (1) c dng:

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16

2

42 6 8 0 2 4 2

1x

tt t x

t loai


a. (HDB 2006)2 21 29 10.3 1 0x x x x

b. 2 8 53 4.3 27 0x x c. 2 23 3 24x x d. 2 22 1 17.2 20.2 12 0

x x Gii:

a. Pt2 21 10

9 .3 1 09 9

x x x x 2 22

3 10.3 9 0x x x x

t2

3 , 0x xt t

Pt 21

10 9 09

tt t

t

Vi t = 12 2 0 2 03 1 3 3 0

1

x x x xx

x x

x

Vi t = 92 2 2 2 2 13 9 3 3 2 2 0

2x x x x

xx x x x

x

b. 8 2 53 .3 4.3 .3 27 0x x 2

6561. 3 972.3 27 0x x (*)

t 3 0xt . Pt (*) 2

1

96561 972 27 01

27

t

t t

t

Vi 21

3 3 29xt x

Vi 31

3 3 327

xt x

Vy phng trnh c nghim: 2, 3x x

c. 22 2 93 3 24 9.3 24 0 9. 3 24.3 9 0

3x x x x x

x

(*)

t 3 0xt

Pt (*) 23

9t 24 9 0 1( loai)

3

t

tt

Vi 3 3 3 1xt x Vy phng trnh c nghim: 1x

d. t2 12xt , v

22 1 11 1 2 2 2xx t Khi pt c dng:

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17

22 1 2

27 20 12 0 2 2 1 2 06

7

x

t

t t x xt loai


a. 6.2 2 1x x

b. 64.9 84.2 27.6 0x x x c. 4 2 13 4.3 27 0x x d. 2 125 10 2x x x Gii:

a. Pt1

6. 2 12

x

x . t xt 2 , t 0

Pt 2 21

3 ( )16. 1 6 6 0

2 2 2 1xt

t t t t t t t x

loai

b. PT

2

4 1623 94 4

64.9 84.2 27.6 0 27. 84. 64 0 13 3 4 4

3 3

x

x x

x x x

x

x

x

c. 22 24 2 13 - 4.3 27 0 3 12.3 27 0x xx x

t 23 ; 0xt t ta c 2 12 27 0t t

2

2 2

13 3 3 2 1

29 2 23 9 3 1

x

x

t x x

t xx

d. 2 25 2.5 2.2xx x

Chia hai v ca phng trnh cho 22 0x , ta c:

2

5 52

2 2

x x

t5

2

x

t

, iu kin 0t

Khi pt (*) c dng:

2

1 52 0 1 0

2 2

xtt t x

t l

Bi 8: Gii ccphng trnha. 9 9 3log log log 274 6.2 2 0x x

b. (H D 2003) 22 2

2 2 3x x x x Gii:

a. Pt 39

9 3log log log 322 6.2 2 0

xx

log99

2log 32 6.2 2 0

xx

t 9log2 xt , t 0 .

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18

Pt 22

6 8 04

tt t

t

Vi t = 2 9 9log log 1 92 2 2 2 log 1 9x x

x x

Vi t = 4 9 9log log 2 292 4 2 2 log 2 9 81x x

x x

b. 22 2

2 2 3x x x x 2

2

42 3

2

x x

x x

t 2

2 0x xt t ta c 2 13 4 0

4

t loait t

t

22 4x x 2 2 0x x

1

2

x

x

Bi 9: Gii ccphng trnha. 3 3 3log log log 94 5.2 2 0x x b. 3.16 2.81 5.36x x x

Gii:a. Pt

233

log log 3log22 5.2 2 0x

x log3

32

log 22 5.2 2 0x

x

t 3log2 xt , 0t .

Pt 21

5 4 04

tt t

t

Vi t = 1 3 3log log 0 32 1 2 2 log 0 1x x

x x

Vi t = 4 3 3log log 2 232 4 2 2 log 2 3 9x x

x x

b. Chia c hai v cho 36x ta c

PT 16 81 4 93. 2. 5 3. 2. 5 036 36 9 4

x x x x

t4

( 0)9

x

t t

Khi phng trnh tng ng21 13 5 23. 2. 5 0 0

20 0 3

tt tt

t tt

t t

Vi4

1 1 09

x

t x

Vi2 4 2 1

3 9 3 2

x

t x

Vy phng trnh c 2 nghim phn bit 0x hoc1

2x


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19

a. 3 32( log 2) log 23 2 3x x b. (HDB 2007) 3x 1 2x x2 7.2 7.2 2 0 Gii:

a. Pt 3 32( log 2) log 23 3 2 0x x . t t =

3log 23x , 0t .

Pt 21( )

2 02

tt t

t

loai

Vi t = 2 3log 2 3 33 2 log 2 log 2 0x

x x

b. 3 22 7 7 2 0 ( 2 , 0)xt t t t t

2( 1)(2 5 2) 0t t t 1

1 22

t t t

0 1 1x x x

Bi 11: Gii phng trnh2

51 2 94

x

x

Gii:

Pt2

52

12 9

2

x

x

22 5 2( 2) 5 4 2 5

4 5

2 2

2 2 9 2 2 9 2 2 9 0

2 2 16 329 0 9 0

2 2 22

xx x x x x

x x xx

t xt 2 , 0t .

Pt 216 32 9 0t t

2

2216 32 9 0 9 32 16 0t t t tt

2

4

4 42 2 log 9

9 9x

t

t x

=


a.2

2

9 10 4

2 4

x

x

b.

27 278 9.2 64

8 2x x

x x

Gii:

Pt 2 29.4 2 . 10 4x

x

2 2 2 2 2 22 2

36 2 .10 2 . 2 10. .2 362 2

x x xx x x

t t = 2x, 0t .

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20

Pt 2 10 144 0t t 8

18( )

t

t loai

x x 32 = 8 2 = 2 x = 3

2

2 2210.236 10.2 2 36.4 2 10.2 144 0

4 4

xxx x x x

b. Phng trnh: 27 278 9.2 648 2

x x

x x

3

2

02 13 32 64 2 4 4 4.2 3 0

log 32 2 2 3

x

x x x x

x x x

x

x


a. 23

2. 0,3 3100

xx

x b.

276. 0,7 7

100

xx

x

Gii:

a. Pt

2

23 32. 31010

xx

x

222

2

3 3 3 3 3 32. 3 0 2. 3 0 2. 3 0

10 10 10 10 1010

x x x x xx

x

t3

10

x

t

, 0t .

Pt 2 2 3 0t t

3

1( )

t

t loai

x

3

10

3= 3 x = log 3

10

b. Bin i phng trnh v dng:

2

7 76. 7 1

10 10

x x

t7

10

x

t

, iu kin 0t

Khi pt (1) c dng:

2

710

7 7

6 7 0 7 log 71 10

xt

t t xt l

Bi 14: Gii ccphng trnha. 8 18 2.27x x x b. (H A 2006)3.8 4.12 18 2.27 0x x x x Gii:a. Chia hai v pt cho 27x , ta c :

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21

Pt8 18

227 27

x x

x x

33

3

3

8 18 2 2 2 22 2 0 2 0

27 27 3 3 33

2 22 0

3 3

xx x x x x

x x

t2

3

x

t

, 0t .

Pt 3 2 0t t 0

2 2 21 1 0

3 3 3

x x

t x

b. 3 2 2 33.2 4.3 2 3 2 2.3 0x x x x x x

Chia 2 v ca Pt cho3x

3 ta c:

3 22 2 2

3. 4 2 03 3 3

x x x

t2

3

x

t

, 0t ta c: 3 23 4 2 0t t t 1

2

3

t

t

Do K ta ch nhn2 3 3

13 2 2

x

t x


a. (H L 2001)2

222 4log6log2log 3.24 xx x b. 2 2log log 626.9 6 13.x x x

Gii:a. iu kin: x > 0.

Ta c: 2 2 2log 2 1 log log4 4 4.4x x x ; 2 2

log 6 log6 xx v2

2 2 2log 4 2 2 log log3 3 9.9x x x

Do phng trnh tr thnh:

2 22 2 2

log log3 9log log log4.4 6 18.9 4 18.

2 4

x xx x x

(*)

t2log3

2

x

t

. iu kin: t > 0.

Khi phng trnh (*) tr thnh 4 t = 18t2 218 4 0t t

4

91

( ).2

t

t lo ai

Vy phng trnh2

2

log3 4 log2

2 9

xx

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22

Vy1

4x l nghim ca phng trnh.

b. iu kin x 0

Cch 1: Ch cng thc: log logb bc aa c vi a, b, c 0 v 1b

p dng cng thc trn, ta chuyn phng trnhlog 62

2log 26.9 6 13.x x x v phng trnh:2 2log log26.9 6 13.6x xx

t 22log 2 4t t

t x x x

Khi ta c phng trnh: 6.9 6.4 13.6t t t Cch 2: Ta c: 2 2log log 626.9 6 13x x x

2 2 2 2 2 2log log 4 log 6 log log log6.9 6 13 6.9 64 136x x x xx x ... T gii


Bi 1: Gii cc phng trnh saua.

2 222 2 3x x x x b. 9 6 2.4x x x

c.2 25 1 54 12.2 8 0x x x x d. 2 5 13 36.3 9 0x x

e.2 22 2 13 28.3 9 0x x x x f. (HH D 2001) 112.3 3.15 5 20x x x

HD:

a. t2

2 ( 0)x x t t ta c4 14

31 ( ) 2

t xt

t loai xt

b. Chia c hai v phng trnh cho 4x ta c2

3 32 0 0

2 2

x x

x

c. t2

25

2

32 5 12 ( 0) 9

4 5 2 4

x x

xt x x

t tt xx x

d. 1 2x x e. 2 1x x Bi 2: Gii cc phng trnh sau

a. (HL 1998) sin sin

7 4 3 7 4 3 4x x

s: x k k

b. (HNN 1998) 2 3 7 4 3 2 3 4 2 3x x

s: 0 2x x

c. x x

6- 35 6 35 12

d. 7 5 2 ( 2 5) 3 2 2 3 1 2 1 2 0x x x

HD: t (1 2) ; 0xt t

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23

3 2 2( 2 5) 3 1 2 0 ( 1)( ( 2 4) 2 1) 0

1 0

3 2 2 2

11 2

t t t t t t

t x

t x

xt

e. 2 3 2 3 4x x

HD: t 2 3 0x

t t 2 3 21

422 3

t xt

xt t

Bi 3: Gii cc phng trnh saua. (HTCKT 1999) 1 1 24 2 2 12x x x s: 0x

b. (HAN D 1999)2 2sin cos9 9 10x x

2

x k k

c. (HH A 2001) 2 1 -1 15.3 7.3 1 6.3 9 0x x x x

s: 3 33 1

log log5 5

x x

d 2 1 2 2( 1)3 3 1 6.3 3x x x x

s: 311

log 23

x

Bi 3: Gii cc phng trnh saua. (HHP 2000) 25 15 2.9x x x

s: 0x b. (HTL 2000)

2 22 1 2 22 9.2 2 0x x x x s: 1 2x x

c. (HHH 1999) 24.3 9.2 5.6x

x x s: 4x

d.2 2 22 6 9 3 5 2 6 93 4.15 3.5x x x x x x

s: 1 4x x


I. Phng php:

Phng php dng n ph dng 2 l vic s dng 1 n ph chuyn phng trnh ban u thnh 1 phng trnhvi 1 n ph nhng cc h s vn cn cha x.Phng php ny thng s dng i vi nhng phng trnh khi la chn n ph cho 1 biu thc th cc biuthc cn li khng biu din c trit qua n ph hoc nu biu din c th cng thc biu din liqu phc tp.

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24

Khi thng ta c 1 phng trnh bc 2 theo n ph (hoc vn theo n x) c bit s l mt s chnhphng.

II. Bi tp p dng:

Bi 1: Gii phng trnh 23 2 9 .3 9.2 0x x x x Gii:t 3xt , iu kin 0t . Khi phng trnh tng ng vi:

2 22

92 9 9.2 0; 2 9 4.9.2 2 9

2x x x x x

x

tt t

t

Khi :+ Vi 9 3 9 2xt x

+ Vi3

2 3 2 1 02

x

x x xt x

Vy phng trnh c 2 nghim2

0

x

x

Bi 2: Gii phng trnh 2 22 29 3 3 2 2 0x xx x

Gii:

t2

3xt iu kin 1t v22 00 3 3 1xx

Khi phng trnh tng ng vi: 2 2 23 2 2 0t x t x

2 22 2 2

2

23 4 2 2 1

1

tx x x

t x

Khi :

+ Vi2 2

3 32 3 2 log 2 log 2x

t x x

+ Vi22 21 3 1xt x x ta c nhn xt:

2

2

1 1 3 10

1 1 1 1

xVT VT x

VP VP x

Vy phng trnh c 3 nghim 3log 2; 0x x

Bi 3: Gii phng trnh: 9 12 .3 11 0x xx x

Gii:PT

23 12 3 11 0x xx x

t 3 0xt t

xx

x

113

13

(*)0113)(

0

xxf

x

x(a + b + c = 0)

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25

Xt phng trnh (*) ta c

(*)0)2(

,013ln3)('

f

xxfx

c nghim duy nht x = 2

Vy, tp nghim ca phng trnh: S = {0 ; 2}

Bi 4: Gii phng trnh: 2 23.25 3 10 5 3x xx x

Gii:PT 2 23.25 3 10 5 3x xx x

2 2 2 25 3.5 1 3.5 1 3 3.5 1 0x x x xx

2

2 2

2

3.5 1 0 13.5 1 5 3 0

5 3 0 2

x

x x

xx

x

PT 2 5 51 1

1 5 2 log 2 log 33 3

xx

PT 22 5 3x x V tri l hm ng bin v phi l hm nghch bin m (2) c nghim x = 2 nn l nghim duy nht.Vy Pt c nghim l: 52 log 3x hoc x = 2

Bi 5: Gii phng trnh: 2 3 1 34 2 2 16 0 1x x x

Gii :t 2xt , iu kin 0t Khi pt (1) tng ng vi:

4 3 2 4 32 8 16 0 4 2 .4 2 0t t t t t t t u = 4, ta c: 2 4 32 . 2 0u t u t t

22

2

2

1 42 4 0

1 4 2

1 52 5 1 log 5 1

1 5

x

u t t t tt t

u t t t t t

tx

t

Bi 6: Gii phng trnh: 9 2 2 .3 2 5 0 1x xx x

Gii:t 3xt , iu kin 0t Khi pt (1) tng ng vi:

2 2 2 2 5 0t x t x 1 3 5 2 25 2

xt l xt x

Ta on c nghim x = 1V tri (2) l mt hm s ng bin cn v phi (2) l mt hm nghch binVy x = 1 l nghim duy nht ca pt (2)

Bi 7: Gii phng trnh: 23 3 5 5 1x x

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27

2' 1 00

20 00 10

1 0

1 0 1 0

m

Sm

mP

fmm

m

Vy vi 0 1m phng trnh c ba nghim phn bit.

Bi 9: Gii pt 1 13.9 (3 7).3 2 0x xx x (1)

Gii:t 13 , 0xt t .

Phng trnh (1) 23. (3 7). 2 0t x t x 2 2 2(3 7) 12(2 ) 9 30 25 (3 5)x x x x x

3 7 3 5 1

6 33 7 3 5

26

x xt

x xt x

11 13 03 3

xt x

12 3 2xt x x (*)Ta thy 1x l mt nghim ca phng trnh (*)

t :1( ) 3

( ) 2

xf x

g x x

Ta c :1

'( ) 3 .ln 3 0x

f x x R

Suy ra 1( ) 3xf x l hm ng bin trn R v '( ) 1 0g x x R . Suy ra ( )g x l hm nghch bin trn RVy phng trnh (*) c nghim duy nht l 1x .Vy pt (1) c 2 nghim l 0; 1x x .


I. Phng php:

Phng php dng n ph dng 3 s dng 2 n ph cho 2 biu thc m trong phng trnh v kho lo bin iphng trnh thnh phng trnh tch.

II. Bi tp p dng:

Bi 1: (HVQHQT D 1997) Gii phng trnh2 2 23 2 6 5 2 3 74 4 4 1x x x x x x

Gii:

Vit li phng trnh di dng:2 2 2 23 2 2 6 5 3 2 2 6 54 4 4 .4 1x x x x x x x x

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28

t

2

2

3 2

2 6 5

4, , 0

4

x x

x x

uu v

v

Khi phng trnh tng ng vi:

1 1 1 0u v uv u v

2

2

3 2 2

22 6 5

1

1 4 1 3 2 0 2

1 12 6 54 15

x x

x x

x

u x x x

v xx x

x

Vy phng trnh c 4 nghim.

Bi 2: Cho phng trnh:2 25 6 1 6 5.2 2 2.2 (1)x x x xm m

a. Gii phng trnh vi m = 1b. Tm m phng trnh c 4 nghim phn bit.Gii:Vit li phng trnh di dng:

2 22 2 2 2

2 2 2 2

( 5 6) 15 6 1 7 5 5 6 1

5 6 1 5 6 1

.2 2 2 .2 2 2

.2 2 2 .2

x x xx x x x x x x

x x x x x x

m m m m

m m

t:

2

2

5 6

1

2, , 0

2

x x

x

uu v

v

. Khi phng trnh tng ng vi:

2

2

2

5 6

11

31 2 1

1 0 22

2 (*)

x x

x

x

xu

mu v uv m u v m xv m m

m

Vy vi mi m phng trnh lun c 2 nghim x = 3, x = 2

a. Vi m = 1, phng trnh (*) c dng:21 2 22 1 1 0 1 1x x x x

Vy vi m = 1,phng trnh c 4 nghim phn bit: x = 3, x = 2, x = 1b. (1) c 4 nghim phn bit (*) c 2 nghim phn bit khc 2 v 3.

(*)2 2

2 2

0 0

1 log 1 log

m m

x m x m

. Khi iu kin l:

22

2

00 2

1 log 0 1 11 0;2 \ ;1 log 4 8 256811 log 9

256

m

m m

m mmm

mm

Vy vi 1 1

0;2 \ ;8 256

m

tho mn iu kin u bi.

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29

Bi 3: (H D 2006) Gii phng trnh2 2 22 4.2 2 4 0x x x x x

Gii:

t

2

2

2

2

x x

x x

u

v

Suy ra 2. 2 xu v 0, 0u v

Phng trnh thnh:4 4 0 (1 ) 4(1 ) 0 ( 4)(1 ) 0u v uv u v v u v

1v 20

01

xx x

x

Ch :C th bin i tng ng a v phng trnh tch

2 2 2 2

2

2 2

2

2 4.2 2 4 0 2 2 1 4 2 1 0

2 1 2 4 0

x x x x x x x x x x

x x x

Bi 4: Gii phng trnh

a. 2 3 3 1 42 5.2 2 0x x x x b. 22 13 3 12 2 2 2 xx x x

Gii:

a. Ta c: 2 3 3 1 4 2 3 3 1 22 5.2 2 0 2 5.2 4.2 0x x x x x x x x

t :

3 12 3

2 3 1

22

, 02 2

x

x x

x x x

uvu

u vuv

v

.

Khi ta c phng trnh:1

5 4 0 5 4 0

4

u

u u vu uv v

v v u

v

Vi: 3 11 2 1x xu

v

v 3 14 2 4x xu

v

(gii phng trnh i s tm nghim)

Tp nghim phng trnh: 1; 2S

b. t

222

3 32 1 1

2

u f x x xu v x x x

v g x x

2

2 2.2 2 2 2 2.2 2 2 .2

2 2 1 13 3 12 2 1 2 0

0 22 02 1

u v u v u v u v

u

u v

v

u xx x

v xx

Bi 5: Gii phng trnh:

a. 22 2log log

3 1 . 3 1 1x x

x x b.2 25 6 1 6 52 2 2.2 1x x x x

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Gii:a. iu kin: 0x

t 2log

3 1x

u , 2log

3 1x

v

Ta c pt

2 2 2 2 2 11 1 1 0 ... 11uu uv u v uv u xuv

b. Vit li phng trnh di dng:2 25 6 1 7 52 2 2 1x x x x

2 22 2 2 2 2 25 3 15 6 1 5 6 1 5 3 12 2 2 1 2 2 2 .2 1x x xx x x x x x x x x

t

2

2

5 6

1

2, , 0

2

x x

x

uu v

v

Khi , pt tng ng vi:

2

2

5 6 2

21

11 1 1 0

1

32 1 5 6 0

21 12 1 1

x x

x

uu v uv u v

v

xx x

xx

x


a. 2 22

32 2 129 3 3 1

x xxx

b.

22 2 114 2 2 1xx x x

c. 8.3 3.2 24 6x x x d.2 2 22 5 2 4 8 3 6 13 52 2 1 2x x x x x x

Gii:

a. t

2

2

32

2

9 , 03

x x

x

u uv

v

Nhn xt rng: 22

22

2 2

33 2 2222

2 14 39 3 3 33 3

x xx x

xx x

x x

u

v

Khi , pt tng ng vi:

222 2

2 2

332 22 2

22

20

1 1 01

14 3 0

9 3 3 3 303 1 3 3 0

x xx xx x

x x

u vuu v u v v

vv

xx x

xx

x

b. t

2

21

4, 0

2

x x

x

uuv

v

Nhn xt rng: 2 22 2 22 11 1. 4 .2 2 .2 2

x x xx x x xu v

Khi , pt tng ng vi:

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31

2

2

2

21

11 1 1 0

1

04 1 0

1

1 02 1 1

x x

x

uu v uv u v

v

xx x

x

x x

c. t3

, 02

x

x

uuv

v

Khi , pt tng ng vi:

3

8 3 24 3 8 08

3 3 1

32 8

x

x

uu v uv u v

v

x

x

d. Nhn xt:Phng trnh trn c dng f x g x h xa a h x f x g x

t

0

0

f x

g x

u a

v a

PT 0 1 01

uu v uv u v u a u v

v

M 1 1u v

2

2

2 5 2 2

24 8 3

22 1 2 5 2 0 1

24 8 3 02 13

2

x x

x x

x

x xx

x x

x


I. Phng php:

Phng php dng n ph dng 4 l vic s dng k n ph chuyn phng trnh ban u thnh 1 h phngtrnh vi k n ph.Trong h mi th k 1 th phng trnh nhn c t cc mi lin h gia cc i lng tng ng.Trng hp c bit l vic s dng 1 n ph chuyn phng trnh ban u thnh 1 h phng trnh vi 1 nph v 1 n x, khi ta thc hin theo cc bc:Bc 1: t iu kin c ngha cho cc biu tng trong phng trnh.

Bc 2: Bin i phng trnh v dng: , 0f x x

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32

Bc 3: t y x ta bin i phng trnh thnh h:

; 0

y x

f x y

II. Bi tp p dng:

Bi 1: Gii phng trnh1 1 1

8 2 182 1 2 2 2 2 2

x

x x x x

Gii:

Vit li phng trnh di dng:1 1 1 1

8 1 18

2 1 2 1 2 2 2x x x x

t:1

1

2 1, , 1

2 1

x

x

uu v

v

Nhn xt rng: 1 1 1 1. 2 1 . 2 1 2 2 2x x x xu v u v Phng trnh tng ng vi h:

8 1 18 28 18

99;

8

u vu v

u v u vu v uv u v

u v uv

+ Vi u = v = 2, ta c:1

1

2 1 21

2 1 2

x

xx

+ Vi u = 9 v9

8v , ta c:

1

1

2 1 949

2 18

x

xx

Vy phng trnh cho c cc nghim x = 1 hoc x = 4.Bi 2: Gii phng trnh 22 2 6 6x x Gii:t 2xu , iu kin u 0 . Khi phng trnh thnh: 2 6 6u u

t 6,v u iu kin 26 6v v u Khi phng trnh c chuyn thnh h:

2

2 2

2

6 00

1 06

u v u vu v u v u v u v

u vv u

+ Vi u = v ta c:

2 3

6 0 2 3 82(1)

xu

u u xu

+ Vi u + v + 1 = 0 ta c:

22

1 2121 1 21 125 0 2 log2 21 21

(1)2

x

u

u u x

u

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33

Vy phng trnh c 2 nghim lx = 8 hoc 221 1

log .2

x


a. 3 18 1 2 2 1x x b. 23 3 5 5x x

Gii:a. t 3 12 0; 2 1x xu v .

PT3 3

33 2 2

01 2 1 2

2 1 01 2 ( )( 2) 0

u vu v u v

u uv u u v u uv v

2

0

1 5log

2

x

x

b. t 3xu , iu kin 0u Khi , pt (1) tng ng vi:

2 5 5 2u u

t 5v u , iu kin 25 5v v u Khi , pt (2) tng ng vi h:

2

2 2

2

51 0

1 05

u v u vu v u v u v u v

u vv u

TH 1: Vi u v , ta c:

23

1 211 21 1 2125 0 3 log

2 21 21

2

x

u

u u x

u loai

TH 2 : Vi 1 0u v , ta c :

23

1 1717 1 17 124 0 3 log

2 21 17

2

x

u

u u x

u loai

Bi 4: Gii phng trnh: 3 127 2 3 3 2 1x x

Gii :t 3xu , iu kin u >0Khi , pt (1) tng ng vi:

3 3

2 3 3 2 2u u t 3 3 2v u , 3 3 2v u Khi , pt (2) tng ng vi h:

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3 33 3 2 2

3 3

2 2

2 3 3 2 33 3 0

3 2 4 2 3

0

3 0

u v u vu v u v u v u uv v

v u v u

u vu v

u uv v VN

Thay u = v vo (3), ta c:

3 2

2

3 2 0 1 2 0

11 03 1 0

22 0x

u u u u u

uux

u lu u

BI TON 7: S DNG TNH CHT N IU CA HM S

I. Phng php:

S dng cc tnh cht ca hm s gii phng trnh l dng ton kh quen thuc. Ta c 3 hng p dng: Hng1: Thc hin cc bc sau:

Bc 1: Chuyn phng trnh v dng: f(x) = kBc 2: Xt hm s y = f(x). Dng lp lun khng nh hm s n iu (gi s ng bin)Bc 3: Nhn xt:

+ Vi 0 0x x f x f x k do 0x x l nghim

+ Vi 0x x f x f x k do phng trnh v nghim

+ Vi 0 0x x f x f x k do phng trnh v nghim.

Vy 0x x l nghim duy nht ca phng trnh.

Hng 2: Thc hin theo cc bc:Bc 1: Chuyn phng trnh v dng: f(x) = g(x)Bc 2: Xt hm s y = f(x) v y = g(x). Dng lp lun khng nh hm s y = f(x) l

L ng bin cn hm s y = g(x) l hm hng hoc nghch binXc nh 0x sao cho 0 0f x g x

Bc 3: Vy phng trnh c nghim duy nht 0x x

Hng 3: Thc hin theo cc bc:Bc 1: Chuyn phng trnh v dng: f(u) = f(v) (3)Bc 2: Xt hm s y = f(x). Dng lp lun khng nh hm s n iu (gi s ng bin)Bc 3: Khi : (3) u v vi , fu v D

II. Bi tp p dng:


a. 2log2.3 3xx (1) b. 2 212 2 1x x x x

Gii:a. iu kin x 0 . Bin i phng trnh v dng: 2log2.3 3x x (2)

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Nhn xt rng:+ V phi ca phng trnh l mt hm nghch bin.+ V tri ca phng trnh l mt hm ng bin.Do vy nu phng trnh c nghim th nghim l duy nht.Nhn xt rng x = 1 l nghim ca phng t rnh (2) v 2log2.3 3 1x

Vy x = 1 l nghim duy nht ca phng trnh.b. Ta c:

2 2 21 0 2 1 0 1x x x x x x 2 21 12 2 2 2 0x x x x x x (do hm s ty 2 ng bin).

Suyra:0

0

VT

VP

m VT = VP (Gi thuyt) nn ta c:

2

2

1

1 01

2 2x x x

xx

Tp nghim phng trnh: 1x

Bi 2: Gii phng trnh 23 1

23

1log 3 2 2 2

5

x x

x x

(1)

Gii:

iu kin: 21

3 2 02

xx x

x

t 2 3 2u x x , iu kin 0u suy ra: 2 2 2 23 2 3 1 1x x u x x u

Khi (1) c dng:

21

3

1log 2 2

5

u

u

Xt hm s:

212

3 3

1 1( ) log 2 log 2 .5

5 5

x

f x x x x

+ Min xc nh 0; )D

+ o hm:

21 1.2 .5 .ln 3 0,

2 ln 3 5x

f x x Dx

. Suy ra hm s tng trn D

Mt khc 31

1 log 1 2 .5 2.7

f

Do , phng trnh (2) c vit di dng:

23 5

1 1 3 2 12

f u f u x x x

Vy phng trnh c hai nghim3 5

2x

Bi 3: Cho phng trnh22 2 4 22 2 25 5 2

x mxx mx

x mx m

a. Gii phng trnh vi4

5m

b. Gii v bin lun phng trnhGii:t 2 2 2t x mx phng trnh c dng: 2 25 5 2 2t t mt t m (1)

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Xc nh hm s 5tf t t + Min xc nh D = R+ o hm: 5 .ln 5 1 0,tf x D hm s tng trn D

Vy (1) 22 2 2 2 2 0 2 0f t f t m t t m t m x mx m (2)

a. Vi4

5m ta c: 2 2

28 4

0 5 8 4 0 25 5

5

x

x x x xx

Vy vi4

5m phng trnh c 2nghim

22;

5x x

b. Xt phng trnh (2) ta c: 2' m m + Nu 2' 0 0 0 1m m m . Phng trnh (2) v nghim phng trnh (1) v nghim.+ Nu ' 0 m = 0 hoc m = 1.

vi m = 0 phng trnh c nghim kp x = 0vi m = 1 phng trnh c nghim kp x

0= 1

+ Nu1

' 00

m

m

phng trnh (2) c 2 nghim phn bit 21,2x m m m cng l nghim kp

ca (1)Kt lun:Vi m = 0 phng trnh c nghim kp x = 0Vi m = 1 phng trnh c nghim kp x0 = 1Vi 0 m 1 phng trnh v nghim

Vi m 1 hoc m 0 phng trnh c 2 nghim 21,2x m m m

Bi 4: Gii phng trnh2 23 2 2 2 32 9 6 4 3 5x x x x x xx x

Gii:Phng trnh

2 26 4 2 4 62 3 6 2 3 5x x x x x xx x 2 22 4 6 6 42 3 2 4 6 3x x x x x xx x x

t2

2 3 2 34 6

u u v vu x xu v

v x

Xt hm s /1 1 1

2 2 ln 2 1 ln 03 3 3

t t

t tf t t f t t R

/f t ng bin, m

f u f v u v

Ta c phng trnh:2 2 14 6 5 6 0

6

xx x x x x

x

Vy tp nghim phng trnh: 1;6S Bi 5: Gii ccphng trnh

a.2 8 22 2 8 2x x x x x b. 2 2log 3 log 7 2x x x

Gii:

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a. t:2

28 28

u x xv u x x

v x

Phng trnh trn 2 2 2 2u v u vv u u v f u f v

Xt hm s: 2tf t t , ' 2 ln 2 0tf t t R 'f t ng bin

m =f u f v nn2 28 2 8 0u v x x x x x

4

2

x

x

Vy tp nghim phng trnh: 2;4S

b. 2 2 2log log log2 3 7 2x x x

t 2logt x th pt tr thnh:2 3 1

2 3 7 2 2. 17 7 7

t t t

t t t

Xt hm s

2 3 1 2 2 3 3 1 1

2. ' ln ln 2. ln 07 7 7 7 7 7 7 7 7

t t t t t t

f t f x t

f t l hm gim trn R

li c 1 1f nnpt cho lun c nghim duy nht 21 log 1 2t x x Vy pt cho c nghim duy nht 2x Bi 6: Gii ccphng trnh

a. 9 5 4 2 20x

x x x b. 32 2

3

log3. log 1

xx x x

Gii:

a. PT 2 25 2

3 [( 5) 2 ] 3 ( 5) 2 1

3 3

x x

x x x x x x

(1)

V5 2

0 , 13 3

nn v tri l hm s nghch bin trn

Mt khc: 2 1f nn PT 2 2f x f x .

b. iu kin: 0x t 3log 3

tt x x

Phng trnh tr thnh : 3. 22 2 1 2 23 3 1 3 3 1 3 2

tt t t t

t t t (1)

Xt hm s 3uf u u c '( ) 3 ln 3 1 0uf u u

Suy ra 3u

f u u ng bin trn RPT (1) 2 2( 1) 2 1 2 1f t f t t t t

Vi 1 3t x Bi 7: Gii cc phng trnh saua. 2 3 5x x b. 2 3 5x x x Gii:a. Phng trnh nhn nghim 1x

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2 3 5 2 3 5 0x x x x Xt hm s 2 3 5x xf x (xc nh vi mi x )

Ta c / 2 ln 2 3 ln 3 0x xf x x . Suy ra th hm s f x ct trc honh ti duy nht mt im

Vy phng trnh c nghim duy nht 1x

b. Phng trnh nhn nghim 1x Chia hai v ca phng trnh cho 3x

PT2 5

13 3

x x

t2 5

( ) 1 ( )3 3

x x

f x v g x

C hai hm s u c tp xc nh l R

Ta c / /2 2 5 5

( ) ln 0 ( ) ln 03 3 3 3

x x

f x v g x

Suy ra hm s f x nghch bin v hm s g x ng binDo th ca hai hm s ct nhau ti mt im duy nhtVy phng trnh c duy nht mt nghim 1x

Bi 6: Gii phng trnh: 14 2 2(2 1)sin(2 1) 2 0x x x x y

Gii:

PT 2 2 2 1 sin(2 1) 0 (1)2 1 sin(2 1) cos (2 1) 0

cos(2 1) 0 (2)

x x

x x x

x

yy y

y

T (2) sin(2 1) 1 x y .

- Khi sin(2 1) 1x y , thay vo (1), ta c: 2x = 0 (VN)- Khi sin(2 1) 1 x y , thay vo (1), ta c: 2x = 2 x = 1.

Thay x = 1 vo (1) sin(y +1) = 1 1 ,2

y k k Z

.

Kt lun: Phng trnh c nghim: 1; 1 ,2

k k Z

.

Bi 7: Gii phng trnh 3 4 0x x Gii:Cch 1: Ta c 3 4 0 3 4 (*)x xx x

Ta thy 1x l mt nghim ca phng trnh (*)

t :( ) 3

( ) 4

xf x x

g x

Ta c : '( ) 3 . ln 3 1 >0 xxf x . Suy ra ( ) 3xf x x l hm ng bin trn R.M ( ) 4g x l hm hngVy phng trnh (*) c nghim duy nht l 1x

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Cch 2: 3 4 0 3 4 (*)x xx x Ta thy 1x l mt nghim ca phng trnh (*)

Nu 1x , ta c13 3 3

1

x

x

3 3 1 4x x (v l)

Nu 1x , ta c13 3 3

1

x

x

3 3 1 4x x (v l).

Vy phng trnh (*) c nghim duy nht l 1x .Bi 8: Gii cc phng trnh

a. 22 3 1x

x b. 5log 32 x x

c. 22 5 29x

x x d. 24 9 7x

x Gii:

a. Ta c :2

2 3 1

x

x 2 ( 3) 1

x x 3 1

1 2 2

x x

(*)

Ta thy 2x l mt nghim ca phng trnh (*) t : 3 1( ) 2 2

( ) 1

x x

f x

g x

Ta c :3 3 1 1

'( ) .ln ln 0 x2 2 2 2

x x

f x R

Suy ra 3 1( ) 2 2

x x

f x

l hm nghch bin trn R. M ( ) 1g x l hm hng

Vy phng trnh (*) c nghim duy nht l 2x b. iu kin : x 0 Phng trnh 5 2log 3 logx x

t 2log 2t

t t x

Phng trnh 52 1

log 2 3 2 3 5 3. 13 5

t t

t t tt

Xt hm s

2 1 2 1

3. ' ln 0.4 3. ln 0.2 03 5 3 5

t t t t

f t f x t

Suy ra: f t l hm gim trn R

Mt khc 1 1f nn pt (**) c nghim duy nht 21 log 1 2t x x

c. Chia hai v cho 29x

ta c :2 5

129 29

x x

Ta thy 2x l mt nghim ca phng trnh. chng minh 2x l mt nghim duy nht.

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Nu 2x th :

2

2

2 2 4

2929 29 2 5 4 251

29 2929 295 5 25

2929 29

x

x x

x

pt v nghim khi 2x Nu 2x : cm tng t ta cng c pt v nghim.Vy phng trnh c nghim duy nht 2x

d. PT4 1

4 3 7 1 73 3

x x

x x

t: 4 4 4

' ln 03 3 3

x x

f x f x f x

ng bin trn R

1 1 1

1 7. ' 7. ln 0

3 3 3

x x

g x g x g x

l hm gim trn R

Do th hm s hai hm ch c th ct nhau ti 1 im duy nht 2x .Vy pt c nghim duy nht 2x

Bi 9: Gii phng trnh: 3 .2 3 2 1x xx x Gii:Nhn xt: ta thy pt 3 .2 3 2 1x xx x c hai nghim x = 1.

Vi1

2x khng l nghim ca phng trnh nn

PT2 1

32 1

x x

x

Ta c hm sy = 3x

tng trn Rhm s

2 1

2 1

xy

x

lun gim trn mi khong

1 1; , ;2 2

Vy Phng trnh ch c hai nghim x = 1


Bi 1: Gii cc phng trnh sau:

a. 3 2 ( 3 2) ( 5)x

x x

HD:

3 2 3 2( ) ( ) 1

5 5

3 2 3 2;0 1; ; 1

5 5

x x

u u v v

+ Nu 0 : 0; 1 1x xx u v VT

+ Nu 0 : 1; 0 1x xx u v VT

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Vy pt v nghim.

b. 8 (3 1) 4x x c. 2 5 11 1

2 5 1x x

e ex x

d. 2 1 2 2 1 1 22 3 5 2 3 5x x x x x x

e. 2( 3 2) ( 3 2) 10x

x x f. 2 (2 3) 2(1 2 ) 0x xx x s:

b.1

3x c. 2;4x d. 1x e. 2x f. 0;2x

Bi 2: Gii cc phng trnh sau:

a. (TL 2001)21 22 2 ( 1)x x x x

b. 12 4 1x x x

c. (QHQT 1997) ( 3 2) ( 3 2) ( 5)x x x

d. (SPHN 2001) 3 5 6 2x x x

e. (BCVT 1998) (2 3) (2 3) 4x x x

f. 3 2 2 32 3 .2 (1 3 ).2 2 0x x xx x x x g. (2.3 1) 3 2x xx

h. 38 .2 2 0x xx x s:a. 1x b. 1x c. VN d. 0;1x e. 1x f. 0x

g. 1x h. 2x

BAI TON 8: S DNG BT NG THC

Bi 1: Gii phng trnh

2 2 22 3 2 2 2 1

3 4 5 14x x x x x x

HD:

Cch 1: Phng trnh:2 2 22 3 2 2 2 13 4 5 14x x x x x x

Ta c:

22

22 2 2 2

22

1 22 3 2

1 12 2 1 2 3 2 2 2 1

12 1 0

3 3 3 9

4 4 4 4 3 4 5 14

5 5 5 1

xx x

xx x x x x x x x

xx x

Du = xy ra khi v ch khi: x 1 .

Cch 2: Phng trnh:2 2 22 3 2 2 2 13 4 5 14x x x x x x

2 2 2

1 2 1 1 1 2 13 4 5 1 3 4 5 1 9.3 4.4 5 1x x x t t t t t t Dng o hm ta chng minh phng trnh 9.3 4.4 5 1t t t c t = 0 l nghim duy nht.Vi t = 0 ta suy ra x 1 .Vy tp nghim phng trnh: 1S

Bi 2: Gii phng trnh 1 1 2 32 2 3 2x x Giair:Cch 1: S dng BT Cauchy.

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V11 2

2x

v 1 22 x l cc s dng. Nn p dng BT Cauchy cho 3 s11 2

2x

,11 2

2x

v 1 22 x , ta c:

1 1 1 2 31 12 2 2 3 22 2

x x x

Du = xy ra khi v ch khi: 1 1 2 1 21 12 2 2 22 3x x x x x

Cch 2: t xt 2 , t 0 . Khi ta c phng trnh:3

2

22 3 2t

t

3 32 3 2 2 0t t . Ta c 3 2t l nghim ca phng trnh. p dng lc Horner, ta c:

2 33 2 0 23 2 2 3 2 3 4 0

Khi : 3 23 3 3 32 3 2 2 0 2 2 2 4 0t t t t t

3

2 3 32 1

32 2 4 0t x

t t

Bi 3: Gii phng trnh:

2 1 3 2

23

82 2

log 4 4 4x x

x x

Gii:

Ta c 22 2

34 4 4 2 1 3 3 log 4 4 4 1x x x x x 23

88

log 4 4 4VP

x x

Mt khc theo BT Csi, ta c: 2 1 3 2 2 1 3 2 42 2 2 2 .2 2 2 8Cosi

x x x xVT

Du = xy ra

2 1 3 2

23

2 2 8

88

log 4 4 4

x x

x x

Gii h ta c nghim ca phng trnh l x =1

2

Bi tp t gii:

Bi 1: Gii cc phng trnh saua. 3 2

13 8

3x

xx b. 3 1 3 3 2x x c. 22 1 2 2 2x x x x

d.2 2sin cos8 8 10 cos 2x x y e. sin 1 sin4 2 cos( ) 2 0yx x xy f. 9 3 10 2x x x

g.2 227 (6 4 1).9x xx x h. 2 12 2 3 1xx x x

s:

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a. 0x b. 0 1x c. 1x d. ;2 2

kx y l

e. ; 0x k y

f. 0 1x x g.2 1

0;1; ;3 3

x

h. 1x

BI TON 9: S DNG GI TR LN NHT V NH NHT CA HM S

I. Phng php:Vi phng trnh c cha tham s: f x,m g m . Chng ta thc hin cc bc sau:

Bc 1: Lp lun s nghim ca (1) l s giao im ca th hm s (C): y f x,m v ng thng

d : y g m . Bc 2: Xt hm s y = f(x,m)+ Tm min xc nh D+ Tnh o hm y ri gii phng trnh y= 0

+ Lp bng bin thin ca hm sBc 3: Kt lun:+ Phng trnh c nghim min , ( ) max , ( )f x m g m f x m x D

+ Phng trnh c k nghim phn bit (d) ct (C) ti k im phn bit+ Phng trnh v nghim d C

II. Bi tp p dng:

Bi 1: Cho phng trnh 22 2 2 22 2 23 2 2 2

x xx x x x m

a. Gii phng trnh vi m = 8b. Gii phng trnh vi m = 27c. Tm m phng trnh c nghimGii:

Vit li phng trnh di dng:2 22 2 2 2 23 4 2 2x x x x x x m

S nghim ca phng trnh l s giao im ca th hm s:2 22 2 2 2 23 4 2 2x x x xy x x vi ng thng y = m

Xt hm s2 22 2 2 2 23 4 2 2x x x xy x x xc nh trn D = R

Gii hn: lim y Bng bin thin: v 3 1, 4 1 nn s bin thin ca hm s ph thuc vo s bin thin cca hm s

2 2 2t x x ta c:a. Vi m = 8 phng trnh c nghim duy nht x = 1b. Vi m = 27 phng trnh c 2 nghim phn bit x = 0 v x = 2c. Phng trnh c nghim khi m 8

Bi 2: Vi gi tr no ca m th phng trnh

2 4 34 21 1

5

x x

m m

c 4 nghim phn bit

Gii:

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V 4 2 1 0m m vi mi m do phng trnh tng ng vi:

2 4 215

4 3 log 1x x m m

t 4 215

log 1m m a , khi : 2 4 3x x a

Phng trnh ban u c 4 nghim phn bit phng trnh (1) c 4 nghim phn bit

ng thng y = a ct th hm s 2 4 3y x x ti 4 im phn bit

Xt hm s:2

2

2

4 3 1 34 3

4 3 1 3

x x khix hoacxy x x

x x khi x

o hm:2 4 1 3

'2 4 1 3

x khix hoac xy

x khi x

Bng bin thin:

T , ng thng y = a ct th hm s 2 4 3y x x ti 4 im phn bit

4 2 4 215

10 1 0 log 1 1 1 1 0 1

5a m m m m m

Vy vi 0 1m phng trnh c 4 nghim phn bit.

Bi 3: Gii v bin lun theo m s nghim ca phng trnh 2 3 4 1x xm Gii:

t 2 , 0

x

t t phng trnh c vit di dng:2

2

33 1

1

tt m t m

t

(1)

S nghim ca (1) l s giao im ca th hm s (C):2

3

1

ty

t

vi ng thng d:y = m

Xt hm s:2

3

1

ty

t

xc nh trn 0;D

+ o hm: 2 2

1 3 1' ; ' 0 1 3 0

31 1

ty y t t

t t

+ Gii hn: lim 1y t + Bng bin thin:Bin lun:Vi 1m hoc 10m phng trnh v nghim

Vi 1 3m hoc 10m phng trnh c nghim duy nht

Vi3 10m phng trnh c 2 nghim phn bit

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Bi 4: Gii phng trnh 3 5 2.4x x x HD:Ta c: 0 0x VT VP x l nghim ca phng trnh.

1 1x VT VP x l nghim ca phng trnh.Suy ra: x = 0 v x = 1 l nghim ca phng trnh.

V 4 0x nn ta chia hai v phng trnh cho 4x, ta c: 3 5 24 4

x x

Xt hm s: 3 5

24 4

x x

f x

vi x R

Vy phng trnh 3 5 24 4

x x

(hay phng trnh 3 5 2.4x x x ) chnh l phng trnh honh giao

im ca :C y f x v trc honh 0Ox y

o hm: / 3 3 5 5

ln ln4 4 4 4

x x

f x

/ / 2 23 3 5 5

ln ln 04 4 4 4

x x

f x x R

Suyra: /f x ng bin

Mt khc, ta c:

/

/ /

/

3 5 150 ln ln ln 0

4 4 16 0 1 0 0;13 3 5 5

1 ln ln 04 4 4 4

f

f f x

f

Suy ra phng trnh /f x 0 c nghim thuc 0;1 . M /f x ng bin

Nn /

f x 0 c nghim x0 duy nht thuc 0;1 Bng bin thin.

x 0 x0 1 f/(x) 0 +

f(x)

0f x

Kt lun:Phng trnh f x 0 ch c ti a hai nghim

Suy ra: x = 0 v x = 1 l hai nghim duy nht ca phng trnh.Vy tp nghim phng trnh 0;1S

Bi 5: (HDB 2004) CMRphng trnh sau c nghim duy nht 1 1 ( 0)xx

x x x

HD:

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1 1xx

x x 1ln ln 1

xxx x

( 1) ln ln 1x x x x ( 1) ln ln( 1) 0x x x x t ( ) ( 1) ln ln( 1)f x x x x x

1 1( ) ln ln( 1)

1f x x x

x x

;

2

2 2

1( ) 0

( 1)

x xf x

x x

Suy ra f(x) nghch bin trn R+

M:1 1

lim ( ) lim ln 01 1x x

xf x

x x x

f x 0 vi mi x 0 f(x) ng bin trn R+

0

lim ( )x

f x

; f e e 1 eln e 1 0

Vy c 0x thuc 0;e 0f x 0 v 0x l nghim duy nht.

Bi 6: Gii phng trnh: 4 6 25 2x x x Gii:Phng trnh 4 6 25 2 0.x xf x x

Ta c

2 2 4 .ln 4 6 .ln 6 25 4 . ln 4 6 . ln 6 0x x x xf x f x x R

Suy ra f(x) ng bin /R.

Mt khc f(x) lin tc v 0 ln 4 ln 6 25 0f

2 16.ln 4 36.ln 6 50 0 0f f x c nghim 0 0;2x

Vy 0f x c ti a hai nghim, ta c bng bin thin:

Ta c 0 0 2 0f v f

Vy phng trnh c ng hai nghim 0 2.x x

BI TON 10: A V PHNG TRNH TCH

TQ 1: 0 0ab cd ac bd ab cd ac bd a b c d b c

0

0

b c

b c a d a d

TQ 2: 1

1 1 1 01

uu v uv u v

v

Vi phng trnh m: f x g x h xa a h x f x g x

Bi tp p dng:

x

f(x)

f(x)

- +0 2x0

0- ++-

f(x0)

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Bi 1: Gii phng trnha. 12 6 4.3 3.2x x x b. 15 3.5 3 3x x x Gii:a. PT

12 4.3 3.2 6x x x 4 3 3 2 3 3x x x

4 2 3 3 0x x 4 2 0 2 4 123 3 0 3 3

x x

x x

x

x

b. PT

3 .5 3.5 3 3 0 5 3 3 3 3 0

3 3 5 1 0 3 3 0 1 5 1 0

x x x x x x x

x x x xx x

Gii cc phng trnh sau:1. 8.3x + 3.2x = 24 + 6x x = 1 hoc x = 3

2. 12.3x + 3.15x - 5x + 1 = 203. 2x + 3x = 1 + 6x4. 8 - x.2x + 23 - x - x = 05. 2x+1 x+1 x5 + 7 - 175 - 35 = 0

6.

22 2 114 2 2 1xx x x

7. 2 2 2

3 2 6 5 2 3 74 4 4 1

x x x x x x

8. 2 1 1 15.3 7.3 1 6.3 9x x x x = 0

9.3 2 3 42 1 2 1.2 2 .2 2x xx xx x

10. 2 x-1 x x x x-1

x 3 + x 3 - 2 = 2 2 -3 11. 4sinx - 21 + sinx cos(xy) + y2

12. 2 22 22 x +x 2 x +x1-x 1-x2 + 2 - 2 .2 -1 = 0

BI TON 11: PHNG PHP LNG GIC HA

Bi 1: Gii phng trnh:2 21 1

12 2

x x

a a

a a

vi tham s 0;1a

Gii:2 21 1

12 2

x x

a a

a a

2 21 11

2 2

x x

a a

a a

Chia c hai v ca phng trnh cho21

2

x

a

a

,

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ta c:2

2 2

2 11

1 1

xxa a

a a

. V 0;1a nn tn ti gc 0;2

cho tan

2a

.

Thu c phng trnh:

2

2tan 211 tan

x

2

2

1 tan 21 tan

x

1 sin cosx x

Hm s sin cosx x

y l hm nghch bin v ta c

2 2

(2) sin cos 1f . Vy x =2 l nghim duy nht ca phng trnh.

Bi 2: Cho hai phng trnh: 3 2 2 2 1 3x x

(1) v 2 1 2cos9

x (2)

Gi s x l nghim ca phng trnh G (1) . Chng minh rng, khi

x cng l nghim ca phng trnh (2) .

Gii:

2 13 2 2 2 1 3 2 1 3

2 1

x x x

x

t 2 1 2x

t vi t > 0.

Khi phng trnh (1) tr thnh:

2 31 14 3 4 32 2

t t tt

. Xt

1;1t , t

cos , 0;t ta c

3 1 1 24cos 3cos cos32 2 9 3

k

V 0; nn5 7

; ;9 9 9

suy ra 1 2 35 7

cos ; cos ; cos9 9 9

t t t

R rng phng trnh bc ba c ba nghim nn ta khng xt

nghim 1;1t . Mt khc 2 5cos 09t v 3 7cos 09

t do nghim ca phng trnh (1)

l: 1 cos 9t

2 1 2cos

9

x .

Vy nu x l nghim ca phng trnh (1) th x cng l nghim ca phng trnh (2)

Bi 3: Gii phng trnh: 3 14.3 3 1 9x x x Gii:

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iu kin: 1 9 0 0 9 1 0x x x Bin i phng trnh v dng:

3 24.3 3.3 1 3x x x

Vi iu kin (*) th 0 3 1x

t cos 3xt , vi 0,2

t

Khi pt c dng:

3 2

02

4cos 3cos 1 cos cos3 sin cos2

3 28 22

83 2

2 4 2

t

t t t t t t

ktt t k

tk

t t k t l

Ta c: 2 2 2 2 2 2cos cos 2. 2cos 1 cos cos4 8 8 8 4 8 2

Do : 32 2 2 2

3 cos log8 8 2 2

xt x

Bi 4: Gii phng trnh 2 21 1 2 1 2 1 2 .2x x x Gii:iu kin 2 21 2 0 2 1 0x x x

Nh vy 0 2 1x , t 2 sin , 0;

2

xt t

Khi phng trnh c dng:

2 21 1 sin sin 1 2 1 sin 1 cos 1 2cos sin3 3

2 cos sin sin 2 2 cos 2sin cos 2 cos 1 2 sin 02 2 2 2 2 2

cos 0(1) 12 12 6 2

03 22 1sin

22 2

x

x

t t t t t t

t t t t t t t t

tt

x

xtt

Vy phng trnh c 2 nghim1

0

x

x

Bi tp tng hp t gii:


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a. 11342

xx

x b. sin 1 sin4 2 .cos 2 0yx x xy c.1 11

92

x x

x

d. 1 1 1( 4)3 ( 1) 3 1 3 1x x xx x x e. 1 3 2 4 12 3 2x y x y

f.2 2 4 2 1

3 3 6 7 1 2.3x x

x x

g.

2 22 1 2 1 101

(2 3) (2 3) 10(2 3)x x x x

h. (HVQHQT 1997) ( 3 2) ( 3 2) ( 5)x x x

i. (HQGHN D 1997) 32)125(7)215( xxx

k. (HCT D 2000) 2)625()625( sinsin xx

l. xxx )22()154()154( s:a. 0 2 3x x x b. , 0x k k y c. 3log 2x

d. 1 0;1x

e.

1

2x y

f. 1x

g.lg10(2 3)

1lg(2 3)

x

h. V nghim i.5 21

2

0 log 7x x

k. x k k k. 2x


a.44 xx xx b. 1 2 1 22 2 2 7 7 7x x x x x x c.

3 413 4 1 4

.4 3 2 3

xx

d. 1 2 1 23 3 3 5 5 5x x x x x x e. 161 42.2 xx f. 73 31 3 13 82 x xx x

g. xx 1001,0.1000 h. 2 25 7 5 .35 7 .35 0x x x x i. 421

)1(39

xx

k. (H m- D 2001) 1

2 22 4 2 4 4 4 8x x x x x

HD: iu kin 0x

1

2 4 2 2 4 0xx x

l. 2 1 24 .3 3 2 .3 2 6x x xx x x x m. (HKTHN 1997) 25 2 3 5 2 7 0x xx x

s:

a. 31 256x x b. 27

228log

343x c. 2x

d. 35

31log

43x e.

1

2x f. x

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g.1

12

x x h.1

2x i.

3 1

2 2x x

k.1

2x l. 3

31; ; log 2

2x

m 1x

Bi 3: Gii cc phng trnh saua. 62

6

1

2

12

3

13

x

xx

x

x

x b. 8444242 22 xxxxx

c. (HQGHN 2000) 2 2log log 22 2 2 2 1x x

x x d. 04.66.139.61

.611

xxx

e. 1223

2

1

3229

xxx

x f. 02525 21 xxxx

g. 324log 242 2 xx x h. 3loglog29log 222 3. xxx x

i. 052.2 82 log3log xx xx l. 5log3log 22 xxx m. 329log2 xx n. xxxx 3223 7.955.97 s:c.Bi 4: Gii cc phng trnh saua. 7503333 4321 xxxx b. 3421 5353.7 xxxx

c. 123

694

xxx

d. xxxx 3.25.235 22

e. 2112222

2332 xxxx f. 13250125 xxx

PHNG TRNH M C CHA THAM S

Bi 1: (HDB -2002)Tm a phng trnh sau c nghim 21 1 1 1 29 2 3 2 1 0x xa a Gii:iu kin [-1;1]x

t21 13 xt ; [-1;1] [3;9]x t

Ta c: (1) vit li2

2 2 2 1( 2) 2 1 0 ( 2) 2 12

t tt m t m t m t t m

t

Xt hm s 2 2 1

2

t tf t

t

, vi [3;9]t .

Ta c:

2/ / 14 3

( ) , ( ) 0 3( 2)

tt t

f t f t tt

Lp bng bin thin

t 3 9f/(t) +

f(t)64

7

4

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52

Cn c bng bin thing, (1) c nghim [-1;1]x (2) c nghim [3;9]t 64

47

m

Bi 2:Cho phng trnh 1 12.4 5.2 0x x m (1) vi m l tham s

a. Gii phng trnh ng vi 2m b. Xc nh tt c cc gitr ca tham s m phng trnh (1) c nghimGii:

Cho 1 12.4 5.2 0x x m (1)a. Gii (1) khi 2m

t 12 xt iu kin1

2t (v 1 1x )

Khi (1) tr thnh: 22 5 0t t m (*)

Vi 2m (*) tr thnh: 22 5 2 0t t 1

22

t t

Vy (1) 1 1 12 2 22

x x 1 1 1 2 1 1 4 0x x x x x

b. Tm m (1) c nghim:Ta c: (*) 22 5t t m

Xem hm s: 22 5y t t trn1

[ , )2

,5

' 4 5; ' 04

y t y t

Bng bin thin:

Da vo bng bin thin ta c:

(1) c nghim (*) c nghim trong1

[ , )2

25

8m

Bi tp t gii:

Bi 1: Vi gi tr no ca p th phng trnh .2 2 5x xp c nghim

Bi 2: (HTS 2001) Gii v bin lun phng trnh aaaxx

22Bi 3: (HH 2000) Cho phng trnh 11 4 3 2 2 3 1 0x xk k k

a. Gii phng trnh khi 3k b. Tm tt c cc gi tr ca k phng trnh c hai nghim tri duBi 3: Cho phng trnh 5.16 2.81 .36x x xa a. Gii phng trnh khi 7a b. Tm tt c cc gi tr ca a phng trnh v nghim

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s: a. 32

50 log

2x x b. ;2 10a

Bi 1: Gii v bin lun phng trnh: 0122.52.2 mmm xx

Bi 2: Gii v bin lun phng trnh: 3

25353

xxx

a Bi 3: Xc nh m phng trnh sau c nghim:

2 22 1 12 2 2 1 .2 2 6 0x x

m m m

Bi 4: Tm m phng trnh: 014.1216.3 mmm xx c hai nghim tri du

Bi 5: Cho phng trnh: 022.4 1 mm xx a. Gii phng trnh khi m = 2.b. Tm m phng trnh cho c hai nghim phn bit 1 2x , x sao cho 1 2x x 3

Bi 6: Gii v bin lun phng trnh:

a. 83.3. xx mm

b. 02.2.2 mmm xx

Bi 7: Xc nh m cc phng trnh sau c nghim:a. 0333231 2 mmm xx

b. 0122244 mmm xx

Bi 8: Cho phng trnh: xxxm 36.581.216. a. Gii phng trnh vi m = 3b. Tm m phng trnh c nghim duy nht.

Bi 9: Cho phng trnh: mtgxtgx 223223 a. Gii phng trnh vi m = 6.

b. Tm m phng trnh c ng hai nghim

2

;2

.

Bi 10: Tm m phng trnh sau c nghim duy nht: 123

12

mx

Bi 11: Tm m hai phng trnh sau tng ng:

0439 122

xx

14.2.4 12 xx mm Bi 12: Tm m hai phng trnh sau tng ng:

16224 241 xxx

19.3.9 12 xx mm

Bi 13: Tm m phng trnh sau c nghim duy nht:2

1 3 22x

m

Bi 14: Xc nh m mi nghim ca phng trnh

2 11

1 13 12

3 3

x x

cng l nghim ca bt phng

trnh 2 22 3 6 1 0m x m x m

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54

CHNG II:PHNG PHP GII PHNG TRNH - BT PHNG TRNH- H LGA RIT.

CH 1: PHNG TRNH LGARIT

BI TON 1: S DNG PHNG PHP LGARIT HO V A V CNG C S

I. Phng php:

chuyn n s khi lgarit ngi ta c th lgarit ho theo cng 1 c s c 2 v ca phng trnh, bt phngtrnh. Chng ta lu cc php bin i c bn sau:

Dng 1: Phng trnh:

0 1

log ( ) 0ab

a

f x b f x

f x a

Dng 2: Phng trnh:

0 1log log0a a

af x g xf x g x

Ch :- Vic la chn iu kin 0f x hoc 0g x tu thuc vo phc tp ca f x v g x - Khi c s a l mt hng s tha mn 0 1a th khng cn kim tra iu kin m bin i tng ng lun

II. Bi tp p dng:

Bi 1: Gii phng trnh 29 3 32 log log .log 2 1 1x x x

Gii:

iu kin:

0

2 1 0 0

2 1 1 0

x

x x

x

. Phng trnh c vit di dng:

22

3 3 3 3 3 3

23 3 3 3 3 3

1 12 log log .log 2 1 1 log log .log 2 1 1

2 2

log 2log .log 2 1 1 log 2log 2 1 1 log 0

x x x x x x

x x x x x x

3

3 3

log 0 1

log 2log 2 1 1 0 2 1 2 2 1 1

x x

x x x x x

0

2

11

4 2 1 22 2 1 2x

xx

x xx x

0

2

1 1

44 0x

x x

xx x

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Vy phng trnh c nghim x = 1 hoc x = 4.

Bi 2: Gii phng trnha. 3 4 5log log logx x x b. )12(log1)13(log2 3 55 xx

c. 2 3 4 2 3 4log log log log .log .logx x x x x x d. 2 81 log (5 ) 2log 3 13x x

Gii:a. iu kin x 0 .Ta bin i v cng c s 3:

4 4 3 5 5 3log log 3.log log log 3.logx x v x x

Khi phng trnh c dng:

3 4 3 5 3

3 4 5 3

log log 3.log log 3.log

log 1 log 3 log 3 0 log 0 1

x x x

x x x

Vy phng trnh c nghimx = 1.

b. iu kin .31x

2 2 35 5 5 5

2 3 3 2

2

log (3 1) 1 3log (2 1) log 5(3 1) log (2 1)

5(3 1) (2 1) 8 33 36 4 0

2( 2) (8 1) 0 1

8

x x x x

x x x x x

x

x xx

i chiu vi iu kin ta c .2x c.iu kin 0 *x

Phng trnh2 5 3 5 5 2 3 3 5log 5.log log 5.log log log 3.log .log .logx x x x x x

2

5 2 3 2 3log . log 3. log log 5 log 5 1 0x x

TH 1: 5log 0 1x x tha mn (*)

TH 2: 2 2 3 2 3

3 32 2

log 5 log 5 1 log 5 log 5 1log log

log 3 log 3x x

2 3

2

log 5 log 5 1

log 33x

tha mn (*).

d. iu kin : 3x .Pt 3

1

282

log (5 ) 2log 3 1x x

8 8 81

log (5 ) 2. log 3 1 log (5 )(3 ) 12

(5 )(3 ) 8 1

x x x x

x x x

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Bi 3: (HDB - 2007) Gii phng trnh 4 22 1

1 1log ( 1) log 2

log 4 2xx x

.

Gii:iu kin: x 1

a v 2 22 1

1 1 1 1log ( 1) log ( 2)2 2log 2 2 2x

x x

2 2 2log ( 1) log (2 1) 1 log ( 2)x x x 2 2log ( 1)(2 1) log 2( 2)x x x

22 3 5 0x x 5

12

x x

Do K, ch nhn nghim5

2x

Bi 4: (HDB - 2007) Gii phng trnh 23 3log ( 1) log (2 1) 2x x

Gii:

iu kin: 1x a v 3 32log ( 1) 2 log (2 1) 2x x

3

2

log ( 1)(2 1) 1 ( 1)(2 1) 3

12 3 2 0 2

2

x x x x

x x x x

.

Do K ch nhn x = 2Bi 5: (HDB - 2006) Gii phng trnh 2 2log 2 2log 4 log 8x x x

Gii:

iu kin:1

x 0, x 1, x2

Pt tng ng vi:2 4 8

1 2 1

log log 2 log 2x x x

2 2 2 2 2

1 4 6 1 2

log 1 log 1 log log 1 logx x x x x

2 21 log 2logx x

22 2x x x

Bi 6: (HDB - 2006) Gii phng trnh 31 822

log 1 log (3 ) log ( 1) 0x x x

Gii:iu kin:1 x 3.

Bin i PT

2 2 2log ( 1) log (3 ) log ( 1) 0x x x 2( 1)(3 )

log 01

x x

x

( 1)(3 )1

1

x x

x

2 4 0x x

1 17 1 17

2 2x x

Do K ch nhn1 17

2x

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57

Bi 7: (HDB - 2002) Gii phng trnh2

23

2716log 3log 0

xx

x x

HD:

Vi K:1 1

0, ,3 3

x x x

a v dng 3 3

3 3

8log 3log3 2 log 1 log

x x

x x

Hoc 3log 0 1x x

Hoc3 3

8 3

3 2 log 1 logx x

3

1log

2x 3x

Bi 8: Gii phng trnh

a. 2 3

4 82log 1 2 log 4 log 4 1x x x

b. 2 22 1 4 14 2

log log ( 2 1) log ( 4 4) log ( 1) 0x x x x x x

c. 3 9 27log log log 11x x x

Gii:

a. iu kin:

1 04 4

4 01

4 0

xx

xx

x

22 2 2 2 2

2 22 2

(1) log 1 2 log 4 log 4 log 1 2 log 16

log 4 1 log 16 4 1 16

x x x x x

x x x x

+ Vi 1 4x ta c phng trnh 2 4 12 0 (2)x x ;

2(2)

6

x

x

lo i

+ Vi 4 1x ta c phng trnh 2 4 20 0x x (3)

2 243

2 24

x

x loai

Vy phng trnh cho c hai nghim l 2x hoc 2 1 6x b. iu kin 1; 2x x

2 2 2 2

2 2

log log 1 log 2 log ( 1) 0log log 2

x x x x

x x

do 1x

4| 2 |

1

xx x

x loai

Vy pt c nghim x = 4.c. iu kin : 0x .

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Pt 2 33 3 3log log log 11x x x

3 3 3 3

6

3 3 3

1 1 1 1log log log 11 1 .log 11

2 3 2 3

11 6

log 11 log 11. log 6 3 7296 11

x x x x

x x x x

Bi 9: (HDB 2002)Gii phng trinh 84 221 1log 3 log 1 log 42 4x x x Gii:

2 2 2 2 2

0, 10, 14

log 3 log 1 log (4 ) log 1 log3

x xx x

xx x x x

x

2 2

0, 1 0 1 1 0 1 14 4 4

1 1 1 2 3 4 2 3 43 3 3

x x x xx x

x x xx x x x x x x x x

x x x

2 2

0 1 13 2 3 3

6 3 0 2 3 0

x xx x

x x x x

Bi 10: Gii phng trnh 25 25log ( 4 13 5) log (3 1) 0x x x

Gii:

iu kin:24 13 5 0

3 1 0

x x

x

Pt 25 5log ( 4 13 5) log 3 1x x x 24 13 5 3 1x x x

t 3 1 2 3x y . Ta c h phng trnh2

2

4 13 2 8 04 12 3 8 0

x x y

y y x

Gii h c y x hoc 2 5 2y x

Vi y x 24 15 8 0x x , tm c nghim15 97

8x

Vi 2 5 2y x 24 11 3 0x x , tm c nghim11 73

8x

Vy tp nghim ca pt cho l15 97 11 73

;8 8

T

Ch :

Pt 2 2

2 25 1 5 14 10 3 1 3 1 2 3 14 4 2 2

x x x x x x

Bi 11: Gii phng trnh :2 21 2 2 1 2 2 22

log (5 2 ) log (5 2 ).log (5 2 ) log (2 5) log (2 1).log (5 2 )xx x x x x x

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Gii:

iu kin:1 5

2 20

x

x

.

PT cho tng ng vi22 22 2 2 2

2

log (5 2 )log (5 2 ) 2log (5 2 ) 2 log (5 2 ) log (2 1)

log (2 1)

xx x x x

x

2

2 2

2

1

4log (2 1) 11

log (5 2 ) 2 log (2 1) 22

log (5 2 ) 0 2

x

x

x x x x

xx

Kt hp vi K trn PT cho c 3 nghim1 1

24 2x x x

Bi 12: Gii phng trnh: 1log cos sin log cos cos 2 0xx

x x x x

Gii:

iu kin:

0 1

cos sin 0

cos cos 2 0

x

x x

x x

.

Khi Pt cos 2 sin cos 2 cos2

x x x x

22 222

22 2

6 32

x kx x k

kxx x k

Kt hp vi iu kin ta c:2

6 3

kx

(Vi *k N ).

Bi 14: Gii cc phng trnh:

a. 2

3

1log 3 2 1 2x x x

b. 2 2 4 2 4 22 2 2 2log 1 log 1 log 1 log 1x x x x x x x x

c. 3log 4.16 12 2 1x x

x

Gii:a.

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PT

2

3 33 0 3

2 23 1 2

3 4 3 23 3 13 2 1 3 1 1

x xx x

x xx x

x x x xx xx x x

x x

2 2

2 2

1 11 4 2 1

4 0 2 09 13 0 3 1 0

3 4 3 2

x xx x

x xx x x x

x x x x

1 4 2 1

9 29 3 5 9 29 5 32 2 2 2

9 29 3 5

2 2

x x

x xx x

x x

b. Phng trnh

4 2 4 2 4 22 2 2

4 2 4 22

log 1 log 1 log 1

0log 1 0 0

1

x x x x x x

xx x x x

x

c. PT 2 1 2 24.16 12 3 4.4 4 .3 3.3x x x x x x x .

Chia 2 v cho 23 0x , ta c:2

4 44 3 0

3 3

x x

.

t 4 , 03

x

t t

. PT tr thnh

2

1

4 3 0 3

4

t loai

t tt

Khi3

4t , ta c:

14 3 4

13 4 3

x

x

.


Bi 1: Gii phng trnh: 29 3 3log ( 1) log (4 ) log (4 )x x x

HD:

iu kin 4 41x

x

(*) 2 23 3log 1 log 16 1 16x x x x

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2

2

1 4 1 6115 0 2

4 1 1 69

217 0

x

xx x

xx

x x


a.3

3 2 3 2

3 1log .log log log

23

xx x

x

HD: iu kin 0x

2 3 3

1

log (1 2 log 6log 2) 0 3

8

x

x xx

b. 51

2log( 1) log log2

x x x

HD: iu kin 1x

2 2 1log( 1) logx2

x x (PTVN)

c. 22 2log ( 3) log (6 10) 1 0x x

HD: iu kin 3x

2 22 2

1 ( )log ( 3) log (3 5) 3 3 5

2

x loaix x x x

x

d. 21

log( 10) log 2 log 42

x x

HD: iu kin 10 0x 5

( 10) 255 5 2

xx x

x

e. 2 ( 5)( 2)log ( 3) log ( 3)xx x x x

HD: iu kin 3x TH 1: 3 1 2x x l nghim ca ptTH 2: 3 1 2x x

22

( 3)( 3)

31 12 5

1log ( 5)log ( 2) xx

xx x x

xxx x

Bi 3:a. 4log ( 2).log 2 1xx

HD: iu kin 0 1x

4 2 2 2

1 ( )1 1log ( 2) log log ( 2) log

2log 2 2x

x loaix x x x

x

b. 2 22 2 2log ( 3 2) log ( 7 12) 3 log 3x x x x

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HD: iu kin4

3 2

x

x

2 2( 1)( 2)( 3)( 4) 24 ( 5 4)( 5 6) 24x x x x x x x x

t

2

5 5x x t phng trnh tr thnh 1 1 25 5.t t t Giic

0

5

x

x

c. 2 23 3log ( 2) log 4 4 9x x x

s: 25 29x x

d. (HAN 2001) 2 23

1log 3 1 2 log 1

log 2xx x

s: 1x


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