200 BAI Mu Logarit

7/30/2019 200 BAI Mu Logarit

1/70

PhPhPhPhng tring tring tring trinhnhnhnh

BBBBt pht pht pht phng tring tring tring trinhnhnhnh HHHH phphphphng tring tring tring trinhnhnhnh

HHHH bbbbttttphphphphng tring tring tring trinhnhnhnh

MMMMuuuu &&&& LLLLogaritogaritogaritogaritThs. LThs. LThs. LThs. LVVVVnnnnoaoaoaoannnn

www.VNMATH.com


2/70

Bi1.Bi1.Bi1.Bi1. Cao ng SPhm Tp. H Ch Minh nm 2002Gii cc phng trnh v bt phng trnh sau

1/ ( )5 x2 log x log 125 1 1 <

2/ ( )2 2x x 5 x 1 x 54 12.2 8 0 2 + =

Bi gii tham kho

1/ Gii bt phng trnh : ( )5 x2 log x log 125 1 1 < iu kin : 0 x 1< .

( ) 5 5125 5

1 31 2 log x 1 0 2 log x 1 0

log x log x < <

5 5 52

5

1t log x 0 t log x log x 1x

53 32t t 3t 1 0 t 0 log x0 1 x 5 52 2t

= = <


3/70

Bi3.Bi3.Bi3.Bi3. Cao ng SPhm Nha Trang nm 2002Gii phng trnh : ( ) ( )log23 3x 1 log x 4x x 16 0+ + =

Bi gii tham kho

iu kin : x 0> Tp xc nh ( )D 0;= + .

t3

t log x= v do x 0 x 1 0> + . Lc : ( ) ( ) 2x 1 t 4xt 16 0 + + = .

Lp ( ) ( ) ( ) ( )2 22' 4x 16x 16 4 x 2 4 x 2 2 x 2 , do x 0 = + + = + = + = + > .

( )

( )

2x 2 x 2 4t

x 1 x 12x 2 x 2

t 4x 1

+ + = =+ + + = =

+

.

Vi3

1t 4 log x 4 x

81= = = .

Vi ( )34 4t log x 1x 1 x 1= =

+ +

Nhn thy phng trnh ( )1 c mt nghim l x 3= .

Hm s ( ) 3f x log x := l hm sng bin trn ( )0;+ .

Hm s ( ) 4g xx 1

=+

c ( )( )

( )2

4g ' x 0, x g x :

x 1

= <

+nghch bin trn ( )0;+ .

Vy phng trnh ( )1 c mt nghim duy nht l x 3= .

So vi iu kin, phng trnh c hai nghim l1

x , x 381

= = .

Bi4.Bi4.Bi4.Bi4. Cao ng Kinh T K Thut Hi Dng nm 2002Gii bt phng trnh : ( )

2 2 22 x 1 x 2 x4x x.2 3.2 x .2 8x 12++ + > + +

Bi gii tham kho

( )2 2 22 x x 2 x4x 2x.2 3.2 x .2 8x 12 0 + + >

2 2 2x x 2 2 x2x.2 8x 3.2 12 4x x .2 0 + + >

2 2 2x x 2 x2x 2 4 3 2 4 x 2 4 0 + >

( ) ( ) ( ) ( )2 2x 2 x 22 4 2x 3 x 0 f x 2 4 x 2x 3 0 1

+ > =


4/70

2x2 4 + 0 0 + +

2x 2x 3 + + 0 0 +

( )f x + 0 0 + 0 0 +

Da vo bng xt, tp nghim ca bt phng trnh l : ( ) ( )x 2; 1 2; 3 .Bi5.Bi5.Bi5.Bi5. Cao ng khi T, M nm 2004 i hc Hng Vng

Gii h phng trnh :( ) ( ) ( )

( )

22log 3log xy

2 2

9 3 2. xy 1

x y 3x 3y 6 2

= + + = + +

Bi gii tham kho

iu kin : xy 0> .

( )

( ) ( )( ) ( )

( )

222 2

2

log xylog xy2. log xy log xy

2 log xy

t 3 1 Lt 3 01 3 2.3 3 0

t 2t 3 0 t 3 3

= = = > = = = =

( ) ( ) 2log xy 1 xy 2 3 = = .

( ) ( ) ( ) ( ) ( ) ( )2 2 x y 5

2 x y 3 x y 2xy 6 0 x y 3 x y 10 0 4x y 2

+ = + + = + + = + =

.

( ) ( )

( )

2

xy 25 17 5 17

x xx y 5 y 5 x2 23 , 4

x 5x 2 0xy 2 5 17 5 17y yVNx y 2 2 2

= + = = + = = + == + = = + =

.

Bi6.Bi6.Bi6.Bi6. Cao ng SPhm Hi Phng i hc Hi Phng nm 20041/ Gii phng trnh : ( ) ( ) ( ) ( )

2

2 1 2

2

1log x 1 log x 4 log 3 x

2 + + =

2/ Gii phng trnh : ( ) ( ) ( )2 23 2log x 2x 1 log x 2x+ + = + Bi gii tham kho

1/ Gii phng trnh : ( ) ( ) ( ) ( )2

2 1 2

2

1 log x 1 log x 4 log 3 x2

+ + =

iu kin :

x 1 0 x 14 x 3

x 4 0 x 4x 1

3 x 0 x 3

< > >


5/70

2

2

2

x x 12 0

x 1 x x 12

x 1 x x 12

+ = + = +

4 x 3

x 1 14 x 1 14

x 11 x 11

= + = = =

x 11

x 1 14

=

= +

.

Kt hp vi iu kin, nghim ca phng trnh l : x 11 x 1 14= = + .

2/ Gii phng trnh : ( ) ( ) ( )2 23 2log x 2x 1 log x 2x+ + = +

iu kin : ( )( ) ( )

( ) ( )22

2

x 2x 1 0 x 1 0x ; 2 0;

x 2x 0 x ; 2 0;

+ + > + > + + > +

.

t : ( ) ( )2 t

2 23 2 2 t

x 2x 1 3 0log x 2x 1 log x 2x t

x 2x 2 0

+ + = >+ + = + = + = >

( )

( )

2 t2 t 2 t 2 tt t

2 t t t t t

x 2x 2 1x 2x 3 1 x 2x 2 x 2x 2

2 1

x 2x 2 3 1 2 2 1 3 1 23 3

+ = + = + = + = + = = + = + =

.

Nhn thy t 1= l mt nghim ca phng trnh ( )2 .

Xt hm s ( )t t

2 1f t

3 3

= + trn :

( ) ( )t t

2 2 1 1f ' t . ln . ln 0, t f t

3 3 3 3

= + < nghch bin trn .

Do , t 1= l nghim duy nht ca phng trnh ( )2 . Thay t 1= vo ( )2 , ta c : 2 2x 2x 2 x 2x 2 0 x 1 3+ = + = = .

Kt hp vi iu kin, nghim ca phng trnh l x 1 3= .

Bi7.Bi7.Bi7.Bi7. Cao ng SPhm Nh Tr Mu Gio TWI nm 2004Gii bt phng trnh :

( )( )2

x 1

1 1log

4 2>

Bi gii tham kho

iu kin : ( )2

0 x 1 1 x 0,1,2< .

( ) ( )x 1 x 1 x 11 1 1 1

log log log x 12 4 2 4

> >

Nu x 1 1 > th ( )1 x 1 1

x 14 1

x 1x 1 14

> >

(v l) Khng c x tha.

Nu 0 x 1 1< < th

( )31 0 x 1 1 0 xx 1 1 40 x 14 1 54x 10 x 1 1 x 24 4

< < <


6/70

Kt hp vi iu kin, tp nghim ca bt phng trnh l3 5

x 0; ;24 4

.

Bi8.Bi8.Bi8.Bi8. Cao ng SPhm Tp. H Ch Minh nm 2004Gii h phng trnh :

( ) ( )2 2

2

4 2

log x y 5

2 log x log y 4

+ = + =

Bi gii tham kho

iu kin :2 2 x 0x y 0

y 0x 0, y 0

>+ > >> >

.

( )( )

( ) ( )2 22 22 2

2 2 2

x y 32x y 32 x y 2xy 32 x y 64

log x log y 4 log xy 4 xy 16 xy 16

+ =+ = + = + = + = = = =

x y 8 x y 8 x y 4

xy 16 xy 16 x y 4

+ = + = = = = = = =

.

Kt hp vi iu kin, nghim ca h l ( ) ( ){ }S x; y 4; 4= = .

Bi9.Bi9.Bi9.Bi9. Cao ng SPhm Bc Ninh nm 2004

Gii bt phng trnh :

( ) ( )( )

2 3

1 1

2 3

log x 3 log x 3

0x 1

+ +

> +

Bi gii tham kho

iu kin : x 3x 1

>

.

Trng hp 1. Nu x 1 0 3 x 1+ < < < .

( ) ( ) ( )2 3

1 1

2 3

log x 3 log x 3 0 + + <

( ) ( ) 3 23 log x 3 2 log x 3 0 + + <

( ) ( ) 3 2 33 log x 3 2 log 3. log x 3 0 + + <

( ) ( ) 3 2log x 3 . 3 2 log 3 0 + <

( ) ( ) 3 2log x 3 0 Do : 3 2 log 3 0 + > < x 3 1 2 x 1 + > < < tha mn iu kin : 3 x 1 < < .

Trng hp 2. Nu x 1 0 x 1+ > > .

( ) ( ) ( )2 3

1 1

2 3

log x 3 log x 3 0 + + >

( ) ( ) 3 23 log x 3 2 log x 3 0 + + >

( ) ( ) 3 2 33 log x 3 2 log 3. log x 3 0 + + >

( ) ( ) 3 2log x 3 . 3 2 log 3 0 + >

( ) ( ) 3 2log x 3 0 Do : 3 2 log 3 0 + < <

www.VNMATH.com


7/70

x 3 1 x 2 + < < khng tha mn iu kin x 1> .

Vy tp nghim ca bt phng trnh l ( )x 2; 1 .

Bi10.Bi10.Bi10.Bi10. Cao ng SPhm Bnh Phc nm 2004Gii phng trnh : ( ) ( )2 3 22 23x 2x log x 1 log x = +

Bi gii tham kho

iu kin : x 0> .

( ) ( )2

2 3 2 32 2

x 1 1log 3x 2x log x 3x 2x

x x

+ = + =

Ta c 2Csi

2 2

1 1 1 1x 0 : x x. x 2 log x log 2 1

x x x x

> + + + = .

Du " "= xy ra khi v ch khi( )

2x 11

x x 1 x 1x 1 Lx

== = = =

.

Xt hm s 2 3y 3x 2x= trn khong ( )0;+ :2y ' 6x 6x . Cho y ' 0 x 0, x 1= = = = .

M( )( ) ( )0;

f 0 0max y 1

f 1 1 +

= = =

2 3y 3x 2x 1 = . Du " "= xy ra khi x 1= .

Tm li :

( )

( )

( )

2

2 3

2 32

1log x 1 1

x

2x 2x 1 2

1log x 3x 2x

x

+ + =

Du " "= trong

( ) ( )1 , 2 ng thi xy ra

x 1 = l nghim duy nht ca phng trnh.

Bi11.Bi11.Bi11.Bi11. Cao ng SPhm Kom Tum nm 2004Gii phng trnh : ( )5 3 5 3log x. log x log x log x= +

Bi gii tham kho

( )5

5 3 55

log x

log x. log x log x 0log 3 =

5 35

1log x log x 1 0

log 3

=

( ) 5 3 3 3log x log x log 3 log 5 0 =

( ) 5 3 3log x. log x log 15 0 =

5

3 3

log x 0 x 1

log x log 15 0 x 15

= =

= =

.

Bi12.Bi12.Bi12.Bi12. Cao ng Giao Thng nm 2004Gii bt phng trnh : ( )1 x x 1 x8 2 4 2 5 1+ ++ + >

www.VNMATH.com


8/70

Bi gii tham kho

( ) ( )x

2x x x

2

t 2 01 8 2.2 2 5 2.2

8 2t t 5 2.t

= > + > + >

( )

2

22

t 0

t 0 5t

5 2t 0 22 t 4 58 2t t 0 t 4

2 1 t 45t 0t 0 1 t255 2t 0 t

28 2t t 5 2t 17

1 t5

> > > > < + > <

+

Bi gii tham kho

iu kin : 3 3

2 2 2

x 0x 0 x 0x 01log x 3 0 log x log 2 x 2 x8

> > >> +

.

( ) ( )2 22 2 2

2 2

log x 3 log x 2 log x 32 0 0

log x 3 log x 3

+ > >

+ +

t2

t log x= . Khi ( ) ( )( )( )

( )2 t 1 t 3t 2t 3

0 f t 0t 3 t 3

+ > = >

+ +.

Xt du ( )( )( )t 1 t 3

f tt 3

+ =

+:

t 3 1 3 +

( )f t + 0 0 +

Kt hp bng xt du v ( ), ta c :

2

2

1 13 t 1 3 log x 1 x8 2

t 3 log x 3 x 8

< < < < < > >

.

Kt hp vi iu kin, tp nghim ca bt phng trnh l 1 1x ;8 2

.

Bi14.Bi14.Bi14.Bi14. Cao ng CKh Luyn Kim nm 2004

www.VNMATH.com


9/70

Gii phng trnh : ( ) ( ) ( )x 3 x 32 2log 25 1 2 log 5 1+ + = + + Bi gii tham kho

iu kin :( )

x 3 x 3 o

x 3 x 3

25 1 0 25 25x 3 0 x 3

5 1 0 5 1 0 , x

+ +

+ +

> > > > + > + >

.

( ) ( ) ( )x 3 x 3

2 2 2log 25 1 log 4 log 5 1+ + = + +

( ) ( ) x 3 x 3 x 3 x 32 2log 25 1 log 4. 5 1 25 1 4.5 4+ + + + = + = +

( ) ( )

x 32

x 3 x 3x 3

5 1 L5 4.5 5 0 x 3 1 x 2

5 5

++ +

+

= = + = = =

Kt hp vi iu kin, nghim phng trnh l x 2= .

Bi15.Bi15.Bi15.Bi15. Cao ng Ha Cht nm 2004Gii phng trnh :

( ) ( ) ( )

x x 1

2 2log 2 1 .log 2 2 6++ + =

Bi gii tham kho

Tp xc nh : D = .

( ) ( ) ( )x x2 2log 2 1 . log 2. 2 1 6 + + =

( ) ( ) x x2 2log 2 1 . 1 log 2 1 6 0 + + + =

( )

( ) ( )

x2

2

t 0 t 0t log 2 1 0t 2

t 2 t 3 Lt t 6 0t 1 t 6 0

> >= + > =

= = + =+ =

( ) x x x2 2log 2 1 2 2 1 4 2 3 x log 3 + = + = = = . Vy phng trnh c nghim duy nht l

2x log 3= .

Bi16.Bi16.Bi16.Bi16. Cao ng Kinh T K Thut Cng Nghip khi A nm 2004Gii phng trnh : 2x 5 x 13 36.3 9 0+ + + =

Bi gii tham kho

Tp xc nh : D = .

( ) ( )2 x 1 x 127.3 36.3 9 0+ + + = x 1

x 1 x 1

2 x 1 1

t 3 0t 3 0 3 1 x 11 x 227t 36t 9 0 3 3t 1 t3

++ +

+

= > = > = = = + = == =

.

Vy phng trnh c hai nghim x 2= v x 1= .

Bi17.Bi17.Bi17.Bi17. Cao ng Cng Nghip H Ni nm 20041/ Gii phng trnh : ( )

2 2

3

x2 cos sin x

4 2sin x8 8.8 1

+ =

2/ Tm tp xc nh ca hm s : ( )2

22 2

1y 4 log x log 3 x 7x 6 2

x

= + +

www.VNMATH.com


10/70

Bi gii tham kho

1/ Gii phng trnh : ( )2 2

3x

2 cos sin x4 2sin x8 8.8 1

+ =

( )2

3 3 21 cos x sin x 12sin x sin x sin x sin x 2 3 21 8 8 8 8 sin x sin x sin x 2

+ + + + + = = = + +

3 2

t sin x, t 1

t 2t t t 2 0

= = =

(loi).

Vy phng trnh cho v nghim.

2/ Tm tp xc nh ca hm s : ( )2

22 2

1y 4 log x log 3 x 7x 6 2

x

= + +

( ) 2 22 22 y 4 log x log x 3 x 7x 6 = + + .

Hm s xc nh khi v ch khi : 22 22

x 0

log x 4 log x 3 0

x 7x 6 0

> + + 2

x 0

x 1 x 6

1 log x 3

>

0 x 1 x 66 x 8

2 x 8

<

.

Vy tp xc nh ca hm s l D 6; 8 = .

Bi18.Bi18.Bi18.Bi18. Cao ng Ti Chnh K Ton IV nm 2004Gii h phng trnh : ( )( ) ( )

2

xx 5x 4 0 12 x .3 1 2

+ + + lun lun ng

x 4; 1 . Do tp nghim ca bt phng trn l x 4; 1 .

Bi19.Bi19.Bi19.Bi19. Cao ng Y T Ngh An nm 2004

www.VNMATH.com


11/70

Gii phng trnh : ( )3

3 2 3 2

3 x 1log . log x log log x

x 23 = +

Bi gii tham kho

iu kin : x 0> .

( ) ( ) ( )33 3 2 3 3 21 1

log 3 log x . log x log x log 3 log x2 2

= +

( ) 3 2 3 21 1 1

1 log x . log x 3 log x log x2 2 2

= +

2 2 3 3 2

1 1 1log x log x. log x 3 log x log x 0

2 2 2 + =

2 2 3 3

1log x log x. log x 3 log x 0

2 =

2 2 3 3log x 2 log x. log x 6 log x 0 =

22 2 3

2

6.log xlog x 2 log x. log x 0log 3

=

2 3 3log x. 1 2 log x 6 log 2 0 =

2

3 3 3 3 3

log x 0 x 1

1 3 3log x 3 log 2 log 3 log 8 log x

2 8 8

= = = = = =

.

Kt hp vi iu kin, nghim ca phng trnh l3

x 1, x8

= = .

Bi20.Bi20.Bi20.Bi20. Cao ng Kinh T K Thut Cng Nghip I nm 2006Gii phng trnh : ( )x 2

5 12xlog 4.log 2

12x 8

=

Bi gii tham kho

iu kin :0 x 1 0 x 1

5 12x 5 20 x

12x 8 12 3

< < > < )

www.VNMATH.com


12/70

2 2

2x x x x4 2.4 1 0

+ =

22

x xx x 2

2

x 0t 4 0t 4 1 x x 0

x 1t 2t 1 0

= = > = = = = + =

.

Vy phng trnh c hai nghim : x 0, x 2= = .

Bi22.Bi22.Bi22.Bi22. Cao ng Xy Dng s 2 nm 2006Gii h phng trnh : ( )

x x2 2

x 22

2 log y 2 log y 5

4 log y 5

+ + = + =

Bi gii tham kho

iu kin : y 0> .

t x2

u 2 , v log y= = . Lc :

( ) ( )( ) ( ) ( )

( )2

22 2

2 u v 2uv 10u v uv 5u v 2 u v 15 0

u v 5 u v 2uv 5

+ + + = + + = + + + = + = + =

( )

x

o2

x

2

u v 5 u 1 2 1 x 2VN

uv 10 v 2 log y 2 y 4

u v 3 u 2 x 42 2

uv 2 v 1 y 2log y 1

+ = = = = = = = = + = = = = = = = =

.

So vi iu kin, nghim ca h phng trnh l : ( ) ( ) ( ){ }S x; y 2; 4 , 4;2= = .

Bi23.Bi23.Bi23.Bi23. Cao ng Giao Thng Vn Ti III khi A nm 2006Gii phng trnh : ( )x

32

1 89x 253 log

log x 2 2x

+ =

Bi gii tham kho

K : 2

0 x 1

x 10 x 1 50 x 1x 0

589x 25 89x 25 89 x ;0 052 2x 2x 89x89

< < < < > < < +

.

( )2 2

3x x x x x

89x 25 89x 253 log 32 log log x log 32 log

2x 2x

+ = + =

2 23 3 4 2

x x

89x 25 89x 25log 32x log 32x 64x 89x 25 0

2x 2x

= = + =

2

2

x 1x 1

525

xx 864

= =

= =

.

Kt hp vi iu kin, nghim ca phng trnh l :5

x8

= .

BBBBiiii22224444.... Cao ng Kinh Ti Ngoi khi A, D nm 2006

www.VNMATH.com


13/70

1/ Gii phng trnh : ( ) ( )2

2 ln x ln 2x 3 0 1+ = .

2/ Gii bt phng trnh :x x

x x

4 2 20

4 2 2

+ >

.

Bi gii tham kho

1/ Gii phng trnh : ( ) ( )2

2 ln x ln 2x 3 0 1+ = .

iu kin :x 0x 0

32x 3 0 x2

> >

.

( )2

2

2x 3 0

2x 3x 1 01 2 ln x 2 ln 2x 3 0 x 2x 3 1

2x 3 0

2x 3x 1 0

= + = =

.

Tp xc nh D = .

( )( )( )( )( )

x x xx

xxx x

2 2 2 1 2 1 x 02 10 0

x 12 22 22 1 2 2

+ < > >> +

.

Vy tp nghim ca bt phng trnh l ( ) ( )x ; 0 1; + .

Bi25.Bi25.Bi25.Bi25. Cao

ng S

Phm H

ng Yn kh

i A n

m 2006

Gii phng trnh : ( ) ( ) ( )x 1 x

2 1 3 2 2 x 1+

+ + =

Bi gii tham kho

Tp xc nh : D = .

( ) ( ) ( )x 1 2x

2 1 2 1 x 1+

+ + =

( ) ( ) ( )x 1 2x

2 1 x 1 2 1 2x 1+

+ + + = + +

( )1 c dng ( ) ( ) ( )f x 1 f 2x 2+ =

Xt hm s ( ) ( )t

f t 2 1 t= + + trn .

www.VNMATH.com


14/70

Ta c ( ) ( ) ( )t

f ' t 2 1 . ln 2 1 1 0= + + + > Hm s ( )f t ng bin trn ( )3 .

T( ) ( ) ( )1 , 2 , 3 x 1 2x x 1 + = = . Vy phng trnh c nghim duy nht l x 1= .

Bi26.Bi26.Bi26.Bi26. Cao ng SPhm Hng Yn khi B nm 2006Gii phng trnh : ( )

5 151 1 1

log sin x log cos x

2 2 25 5 15

+ +

+ =

Bi gii tham kho

iu kin : sin x 0, cos x 0> > .

( ) 5 15log sin x log cos x5 5.5 15.15 5 5. sin x 15.cos x + = + =

3 1 11 sin x 3 cos x cos x sin x cos x cos

2 2 2 6 3

+ = = + =

( ) x k2 x k2 , k6 2

= + = +

.

Kt hp vi iu kin, nghim ca phng trnh l ( )x k2 , k6

= + .

Bi27.Bi27.Bi27.Bi27. Cao ng SPhm Hng Yn khi D1, M nm 2006Gii phng trnh : ( ) ( )9 3log x log 2x 1 1= +

Bi gii tham kho

1/ Gii phng trnh :

( ) ( )

9 3

log x log 2x 1 1= +

iu kin :x 0

x 02x 1 1 0

> > + >

.

( ) ( )3 3log x log 2x 1 1 x 2x 1 1 x 2x 2 2 2x 1 = + = + = + +

2 2x 0

x 2 2 2x 1 x 4x 4 8x 4 x 4x 0x 4

= + = + + + = + = =

.

Kt hp vi iu kin, nghim ca phng trnh l x 4= .

Bi28.Bi28.Bi28.Bi28. Cao ng Bn Cng Hoa Sen khi A nm 2006

Gii h phng trnh :

( ) ( )

2x y2x y

22 23. 7. 6 0

3 3

lg 3x y lg y x 4 lg 2 0

+ = + + =

Bi gii tham kho

iu kin :

x 03x y 0 y

x 0yy x 0 3x 03

> >

> > + > > > .

www.VNMATH.com


15/70


16/70

Do ( )1 lun ng vi x 2 hay ( )x ; 2 2; + l tp nghim ca btphng trnh.

Nu( )

( )

2

2x 4

x 4 2 x 2

2 x 2

3 1x 2 3 x 4 .3 1

x 4 .3 0

= > + >

+ > > < >

x x 113 3 3 x 13

> > > .

Vy tp nghim ca phng trnh l ( )x 1; + .

Bi33.Bi33.Bi33.Bi33. Db Cao ng SPhm H Nam khi A nm 2006Gii phng trnh : ( )

3 3x 5 1 x 5 x x4 2.2 2.4+ + + ++ =

Bi gii tham kho

Tp xc nh : D = .

( )3 3

3 3x 5 1 x 5 x

x 5 x x 5 x

x 2x

4 2.22 0 4.4 2.2 2 0

4 2

+ + + ++ + + = + =

( )

( )

333

3

3

x 5 x 1x 5 x2 x 5 x

x 5 x

2x 5 x

12 t 22 t 0

4.2 2.2 2 0 24t 2t 2 0 2 t 1 L

+ + +

+

+

= = == > + = + = = =

3 23 3x 5 x 1 x 5 x 1 x 5 x 3x 3x 1 + = + = + = +

3 2x 3x 2x 6 0 x 3 + = = .

Vy phng trnh c mt nghim l x 3= .

Bi34.Bi34.Bi34.Bi34. Cao ng K Thut Y T I nm 2006Gii phng trnh : ( ) ( ) ( )x x2 21 log 9 6 log 4.3 6+ =

Bi gii tham kho

iu kin :x

x

9 6 0

4.3 6 0

>

>

.

( ) ( ) ( ) ( ) ( )x x x x2 2 2 2 2log 2 log 9 6 log 4.3 6 log 2. 9 6 log 4.3 6 + = =

www.VNMATH.com


17/70

( ) ( )x2

x x x xx 1

3 1 L2.9 12 4.3 6 2. 3 4.3 6 0 x 1

3 3

= = = ==

.

Thay x 1= vo iu kin v tha iu kin. Vy nghim ca phng trnh l x 1= .

Bi35.Bi35.Bi35.Bi35. Cao ng Ti Chnh Hi Quan khi A nm 2006Gii bt phng trnh : ( )3

3x 5log 1

x 1

<

+

Bi gii tham kho

iu kin :3x 5 5

0 x 1 xx 1 3

> < >

+.

( )3x 5 3x 5 8

3 3 0 0 x 1 0 x 1x 1 x 1 x 1

< < < + > >

+ + +.

Kt hp vi iu kin, tp nghim ca bt phng trnh l5

x ;3

+ .

Bi36.Bi36.Bi36.Bi36. Cao ng K Thut Cao Thng nm 2006Gii phng trnh : ( ) ( ) ( )22 2log x 3 log 6x 10 1 0 + =

Bi gii tham kho

iu kin :2x 3 0 5

x6x 10 0 3

> > >

.

( )

( ) ( )2 22

2 2

2 x 3 2 x 3 x 1

log log 1 1 x 3x 2 0 x 26x 10 6x 10

=

= = + = = .

So vi iu kin, phng trnh c nghim duy nht l x 2= .

Bi37.Bi37.Bi37.Bi37. Cao ng Kinh T Tp. H Ch Minh nm 2006Gii phng trnh : ( )

222 log xx 8

+=

Bi gii tham kho

iu kin : x 0> v x 1 .

( )2 2 22 x 2 x 2

2

12 log x log 8 log x 3. log 2 2 0 log x 3. 2 0log x + = + = + =

32 2 2 2

log x 2 log x 3 log x 0 log x 1 x 2 + = = = .


Bi38.Bi38.Bi38.Bi38. Cao ng in Lc Tp. H Ch Minh nm 2006Gii phng trnh : ( )x 27 3

3log 3 3 log x 2 log x

4 =

Bi gii tham kho

iu kin : 0 x 1< .

( ) 23 3 3 33 3

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

4 log x 4 log x 4 = = =

www.VNMATH.com


18/70

3 3

1 1 1log x log x x 3 x

2 2 3 = = = = .


x 3 x3

= = .

Bi39.Bi39.Bi39.Bi39. Cao ng Kinh T Cng Ngh Tp. H Ch Minh khi A nm 2006Gii bt phng trnh : ( )3

x 2log

x5 1

<

Bi gii tham kho

iu kin :x 2

0 x 0 x 2x

> < > .

( ) 3x 2 x 2 2

log 0 1 0 x 0x x x

< < < > .

Kt hp vi iu kin, tp nghim ca bt phng trnh l ( )x 2; + .

Bi40.Bi40.Bi40.Bi40. Cao ng Kinh T Cng Ngh Tp. H Ch Minh khi D1 nm 2006Gii phng trnh : ( ) ( )1 4

4

1log x 3 1 log

x = +

Bi gii tham kho

iu kin :x 3 0 x 3

x 31 x 00x

> > > >>

.

( ) ( )4 4 41 x 3 x 3 1log x 3 log 1 log 1 x 4x x x 4 = = = = .


Bi41.Bi41.Bi41.Bi41. Cao ng Cng Nghip H Ni nm 2005Gii bt phng trnh :

( ) ( )2

5 5log x log x

5 x 10+

Bi gii tham kho

iu kin : x 0> .

tt

5log x t x 5= = .

( ) ( )2 2tt t t 2

5

15 5 10 5 5 t 1 1 t 1 1 log x 1 x 5

5 +

Kt hp vi iu kin, tp nghim ca bt phng trnh l1

x ;55

.

Bi42.Bi42.Bi42.Bi42. Cao ng Kinh T K Thut Cng Nghip I khi A nm 2005Tm tp xc nh ca hm s : ( )25y log x 5.x 2= + .

Bi gii tham kho

Hm sc xc nh khi v ch khi

www.VNMATH.com


19/70

( )

2

2

2

5

x 5.x 2 0, x 5 1 5 1x 5.x 2 1 x x

2 2log x 5.x 2 0

+ > + + +

.

Vy tp xc nh ca hm s cho l5 1 5 1

D ; ;2 2

+ = +

.

Bi43.Bi43.Bi43.Bi43. Cao ng SPhm C Mau khi B nm 2005Gii phng trnh : ( )

2lg x 2 lg x 3 lg x 2x 10 +=

Bi gii tham kho

iu kin : x 0>

( )2lg x 2 lg x 3 lg x 2 2 2 2lg x lg10 lg x 2 lg x 3 lg x 2 lg x 3 lg x 2 0 + = = + + =

lg x 1 x 10

lg x 2 x 100

= = = =

.

Kt hp vi iu kin, nghim ca phng trnh l x 10 x 100= = .

Bi44.Bi44.Bi44.Bi44. Cao ng SPhm Vnh Phc khi B nm 2006Gii phng trnh : ( )2 20,5 2 xlog x log x log 4x+ =

Bi gii tham kho

iu kin : 0 x 1< .

( )2

2 2 x xlog x 2 log x log 4 log x + = +

22 2

4

1log x 2 log x 1 0log x

+ =

22 2

2

2log x 2 log x 1 0

log x + =

22 2

3 2 2

2

x 2log x 1

t log x t log x 1log x 1 x

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

x4

= = = = = = = = = + = = =

.

So vi iu kin, nghim ca phng trnh l1 1

x x x 24 2

= = = .

Bi45.Bi45.Bi45.Bi45. Cao ng SPhm Vnh Phc khi A nm 2006Gii bt phng trnh : ( ) ( )

xx

4 1

4

3 1 3log 3 1 .log

16 4

Bi gii tham kho

iu kin :x x

3 1 0 3 1 x 0 > .

( ) ( ) ( )x x4 4 43

log 3 1 . log 3 1 log 16 04

+

www.VNMATH.com


20/70

( ) ( ) 2 x x4 43

log 3 1 2 log 3 1 04

+

( ) ( ) ( )

( )

x xx4 44

2x

4

1t log 3 1 log 3 1t log 3 1 x 1

21 3 3 x 34t 8t 3 0 t t log 3 12 2 2

= < = + < > >

.

Kt hp vi iu kin, tp nghim ca bt phng trnh l

( ) ( )x 0;1 3; +.

Bi46.Bi46.Bi46.Bi46. Cao ng SPhm Tp. H Ch Minh khi A nm 2006Gii h phng trnh : ( )2 3

2 3

log x 3 5 log y 5

3 log x 1 log y 1

+ = =

Bi gii tham kho

iu kin :3 3

2 2

x 0, y 0 x 0, y 0 x 0, y 0x 2

5 log y 0 log y 5 y 162

0 y 162log x 1 0 log x 1 x 2

> > > > > >

<

.

t :2

3 32

22

a 5 log y 0 a 5 log y

b log x 1b log x 1 0

= = = =

.

( )2 2

2 2 2 22 2

b 1 3a 5 b 3a 4b 3a a 3b b a 3a 3b 0

3b a 5 1 a 3b 4

+ + = + = + = + + = + = + =

( )( ) ( ) ( )( )a b

b a b a 3 b a 0 b a b a 3 0 a b 3

=

+ = + = + =

( )

( )

23

222

a b a b

a 1 a 4 La 3a 4 0 a 5 log y 1

b 3 ab 3 a b log x 1 1

a 3a 6 0 VNa 9 3a 3

= = = = + = = = = = = = + =+ =

43 3

2 2

5 log y 1 log y 4 y 3 81

log x 1 1 log x 2 x 4

= = = = = = =

.

Kt hp vi iu kin, nghim ca h l ( ) ( ){ }S x; y 4; 81= = .

Bi47.Bi47.Bi47.Bi47. Cao ng SPhm Tp. H Ch Minh nm 2006Gii h phng trnh :

( )( )

x y

5

3 .2 1152

log x y 2

= + =

Bi gii tham kho

iu kin : x y 0+ > .

( )( )

x y x y

x 5 x 5 x

5

y 5 x y 5 x3 .2 1152 3 .2 1152

x y 5 3 .2 1152 2 .6 1152log x y 1

= = = = + = = =+ =

www.VNMATH.com


21/70

x

y 5 x x 2

y 36 36

= = ==

.

So vi iu kin, nghim ca h l ( ) ( ){ }S x; y 2; 3= = .

Bi48.Bi48.Bi48.Bi48. Cao ng Du Lch H Ni khi A nm 2006Gii phng trnh :

( )2

3

log 8 x x 9 2 + + =

Bi gii tham kho

iu kin : 28 x x 9 0 + + > .

( ) 2 2 2 2x 1 0 x 1

8 x x 9 9 x 9 x 1x 4x 9 x 2x 1

+ + + = + = + =+ = + +

x 4 = .

Thay nghim x 4= vo iu kin v tha iu kin. Vy nghim phng trnh l x 4= .

Bi49.Bi49.Bi49.Bi49. Cao ng Kinh T K Thut Ngh An khi A nm 2006Gii phng trnh : ( ) ( )x x 13 3log 3 1 . log 3 3 2++ + =

Bi gii tham kho

Tp xc nh : D = .

( ) ( ) ( ) ( ) ( )x x x x3 3 3 3log 3 1 . log 3. 3 1 2 log 3 1 . 1 log 3 1 2 + + = + + + =

( )( )

( ) ( ) ( )

( )

xxx x333 3

x2

3

log 3 1 1t log 3 1t log 3 1 t log 3 1

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

+ == += + = + = = + = + = + =

( )

xx

3x 2 x

3 23 1 3x log 28

3 1 3 3 L9

=+ = = + = =

.

Vy nghim ca phng trnh l3

x log 2= .

Bi50.Bi50.Bi50.Bi50. Cao ng SPhm Qung Ngi nm 2006Gii phng trnh : ( )x x x8 18 2.27+ =

Bi gii tham kho Tp xc nh D = .

( )

x x2x 3x x

3 2 3 2

3 3t 0 t 03 3 3

1 2. t 12 22 2 2

2t t 1 0 2t t 1 0

= > = > + = = = = =

x 0 = .

Vy phng trnh c mt nghim l x 0= .

Bi51.Bi51.Bi51.Bi51. Cao ng Cng ng H Ty nm 2005Gii bt phng trnh : ( )2x 4 x 2x 23 45.6 9.2 0+ ++

Bi gii tham kho

Tp xc nh D = .

www.VNMATH.com


22/70

( )2x x

x x x 3 381.9 45.6 36.4 0 81. 45. 36 02 2

+ +

xx

2

3 t 0t 0 4 3 4

0 t 02 49 2 91 t

81t 45t 36 0 9

> = > < < > <


23/70

( )( )

2

2

11x 1 2 x 2 1log x 1t

22x 1 4 x 3t 2 log x 1 2

+ + + +

.

Kt hp vi iu kin, tp nghim phng trnh l : ( ) ( ) { }x 1; 2 1 3; \ 0 + .Bi54.Bi54.Bi54.Bi54. Cao ng SPhm Tp. H Ch Minh nm 2005

Gii bt phng trnh : ( ) ( )2xlog 5x 8x 3 2 + > Bi gii tham kho

iu kin : ( )20 x 10 x 1 3

x 0; 1;35x 8x 3 0 5x x 1

5

< < + + > < >

.

( )( ) ( )

2 2

2 2

3x 0;

3 5x 0;5 1 3

x ;5x 8x 3 x 2 2

x 1; x 1;

1 35x 8x 3 xx ; ;

2 2

+ < + + + > +

1 3x ;

2 53

x ;2

+

.

Vy tp nghim ca phng trnh l1 3 3

x ; ;2 5 2

+ .

Bi55.Bi55.Bi55.Bi55. i hc Quc Gia Tp. H Ch Minh khi B nm 2001Gii bt phng trnh : ( ) ( ) ( )21 xlog 1 x 1

Bi gii tham kho

iu kin : 2

2

1 x 01 x 1

1 x 0x 0

1 x 1

> <

Tp xc nh : ( ) { }D 1;1 \ 0= .

( )( )

( )( ) ( ) ( )( )2 2

2 2 2

1 x 1 xlog 1 x log 1 x 1 x 1 1 x 1 x 0

+

( ) 2 2 2x x x 0 x x 0 0 x 1 .

Kt hp vi tp xc nh, tp nghim ca bt phng trnh l : ( )x 0;1 .

Bi56.Bi56.Bi56.Bi56. i hc Quc Gia Tp. H Ch Minh khi A nm 2001Gii phng trnh : ( )

22 2 2log 2x log 6 log 4x4 x 2.3 =

Bi gii tham kho

iu kin :

x 0

x 0x 0

>

> Tp xc nh : ( )D 0;= + .

( ) 2 2 2 2 2 21 log x log x 2 log 2x log x log x 1 log x4 6 2.3 0 4.4 6 2.9 0+ + = =

www.VNMATH.com


24/70

2 2

2 2 2

2log x log x

log x log x log x 3 34.4 6 18.9 0 4 18. 0

2 2

= =

( )

( )

2

2

2

log x2

log x2log x

3 418t t 4 0 t N12 9

log x 2 x3 4t 0 3 1

t L2 2 2

+ = = = = = = > = =

.


x4

= .

Bi57.Bi57.Bi57.Bi57. i hc Ngoi Thng Tp. H Ch Minh khi A nm 2001Gia v bin lun phng trnh : ( )

2 2x 2 mx 2 2x 4mx m 2 25 5 x 2mx m+ + + + + = + +

Bi gii tham kho

t :

2

2

a x 2mx 2

b x 2mx m

= + + = + +

. Lc : ( ) ( )a a b

5 5 b+

= .

Ta c :a a b

a a b

b 0 5 5 0

b 0 5 5 0

+

+

>

. Do : ( ) 2b 0 x 2mx m 0 = + + = .

Lp 2' m m = .

Trng hp 1 : 2' m m 0 0 m 1 : = < < < Phng trnh v nghim.

Trng hp 2 : 2' m m 0 m 0 m 1 : = > < > Phng trnh c 2 nghim phn

bit :2 2

1 2x m m m, x m m m= = + .

Trng hp 3 : 2m 0 :

' m m 0m 1 :

= = = =

Bi58.Bi58.Bi58.Bi58. i hc Y Dc Tp. H Ch Minh nm 2001Cho phng trnh : ( ) ( ) ( )2 2 2 24 1

2

2 log 2x x 2m 4m log x mx 2m 0 + + + = . Xc

nh tham s m phng trnh ( ) c hai nghim 1 2x , x tha :2 21 2

x x 1+ > .

Bi gii tham kho

( ) ( ) ( )2 2 2 22 2log 2x x 2m 4m log x mx 2m + = + 2 2 2 2

2 2 2 2

x mx 2m 0 x mx 2m 0

x 2m x 1 m2x x 2m 4m x mx 2m

+ > + > = = + = +

.

( ) c hai nghim 1 2x , x tha :2 21 2

x x 1+ >

1 2 22 21 2 22 21 1 22 22 2

x 2m, x 1 m m 04m 0

1 m 0x x 1 12m m 1 0 1 m 2 1x mx 2m 0 2 m

5m 2m 0 5 22m 0 mx mx 2m 0

5

= = > < + > < <

+ > < < >+ >

.

Phng trnh c 1 nghim .

Phng trnh c 1 nghim .

www.VNMATH.com


25/70

Vy ( ) 2 1m 1; 0 ;5 2

tha yu cu bi ton.

Bi59.Bi59.Bi59.Bi59. i hc Nng Lm Tp. H Ch Minh nm 2001Tm m bt phng trnh: ( ) ( )2x x x 12 m. log 2 4 x+ + + c nghim.

Bi gii tham kho

iu kin :

x 0

4 x 0 0 x 4

x 12 0

+

Tp xc nh : D 0; 4 = .

Ta c : x 0; 4 th ( )2 2log 2 4 x log 2 1 0+ = > .

Lc : ( )( )2

x x x 12m

log 2 4 x

+ +

+ .

Mt khc : x 0; 4 th( )( ) ( )2

f x x x x 12 :

g x log 2 4 x :

= + + = +

Do :( )( )

f x

g xt min l

( )( )

f 03

g 0= ( )1 c nghim khi v ch khi m 3 .

Bi60.Bi60.Bi60.Bi60. i hc Cn Thnm 2001Xc nh ca mi gi tr ca tham s m h sau 2 nghim phn bit :

( ) ( ) ( )( ) ( )

2

33 32

2 x 2x 5

log x 1 log x 1 log 4 1

log x 2x 5 m log 2 5 2 +

+ > + =

Bi gii tham kho

( ) ( ) ( )3 3 3 3 3

x 1 x 1x 11 x 1 x 12 log x 1 2 log x 1 2 log 2 log log 2 2

x 1 x 1

> > > + + + > > >

x 1

1 x 33 x

0x 1

> <

.

t 2y x 2x 5= + v xt hm 2y x 2x 5= + trn ( )1; 3 .Ta c : y ' 2x 2. Cho y ' 0 x 1= = = .

x 1 3 +

y ' 0 +

y 8

4

Do : ( ) ( )x 1; 3 y 4; 8 .

t min l .

t max l .

www.VNMATH.com


26/70

t ( )22t log x 2x 5= + .

Ta c : ( ) ( ) ( )2 22y x 2x 5 4; 8 t log x 2x 5 2; 3= + = + .

( ) ( ) ( ) ( )2m

2 t 5 f t t 5t m , t 2; 3t

= = = .

Xt hm s ( ) 2f t t 5t= trn khong ( )2; 3 .

( ) ( ) 5f ' t 2t 5. Cho f ' t 0 t2

= = = .

Bng bin thin

t 2 5

2 3

+

( )f ' t 0 +

( )f t 6

6

25

4

Da vo bng bin thin, h c hai nghim phn bit25

m 64

< < .

Bi61.Bi61.Bi61.Bi61. i hc Nng khi A, B t 1 nm 2001Gii h phng trnh :

( )

( ) ( )

x

y

log 6x 4y 2

log 6y 4x 2

+ = + =

Bi gii tham kho

iu kin :x 0, x 1

y 0, y 1

> >

.

( )( )( )

( ) ( )

( ) ( )( ) ( )( )

1 22

2

6x 4y x 12 x y x y x y x y x y 2 0

6y 4x y 2

+ = = + + = + =

2

2

x y x y x y 0

6x 4y x y 0 y 10x y x y 10

y 2 x x 2, y 0y 2 x y 2 x

x 4, y 6x 4 x 26x 4y x

= = = = + = = == = = = = = = = = = = = + =

.

Kt hp vi iu kin, nghim ca h l ( ) ( ){ }S x; y 10;10= = .

Bi62.Bi62.Bi62.Bi62. i hc Nng khi A t 2 nm 2001Tm m bt phng trnh c nghim ng ( ) ( ) 2mx : log x 2x m 1 0 + + >

Bi gii tham kho

( ) ( )2m mlog x 2x m 1 log 1 + + >

www.VNMATH.com


27/70

( )

2 2

2 2

0 m 1

0 m 1 0 m 1 a 1 0 Sai

x 2x m 1 1 x 2x m 0 ' 0m

m 1 m 1 m 1

a 1 0x 2x m 1 1 x 2x m 0

' 1 m 0

< > = > + + > + > = .

Bi63.Bi63.Bi63.Bi63. i hc SPhm Vinh khi A, B nm 2001Gii phng trnh : ( )2 2 24 5 20log x x 1 . log x x 1 log x x 1

+ =

Bi gii tham kho

iu kin :

2

2

2

x x 1 0

x x 1 0 x 1

x 1 0

> + >

.

( ) 2 2 24 20 5 20log 20.log x x 1 . log x x 1 log x x 1 0 + =

2 220 4 5

log x x 1 . log 20.log x x 1 1 0 + =

2220

225 20

4 5 4

x x 1 1log x x 1 0

1log x x 1 log 4log 20.log x x 1 1 0

log 20

= = + + = = + =

202020

22 2

log 4log 42log 42

2 2 2

x 1x 1 0

x 1x 1 x 1x 1 x 2x 1

x 5 ax x 1 5x 1 5 x

x 1 a 2ax x

= = = + =+ + = + = + = +

( ) ( )

20

20

log 42 2

log 4

x 1x 1x 111

x 25 12ax a 1 x a 12a 2.5

== = = = =

.

So vi iu kin, phng trnh c hai nghim l : ( )2020

log 4

log 4

1x 1 x 25 1

2.5= = .

Bi64.Bi64.Bi64.Bi64. i hc Thy Li nm 2001Gii phng trnh : ( ) ( )

2 2x 1 x x2 2 x 1 =

Bi gii tham kho

Tp xc nh : D = .

( ) ( ) ( ) ( )2 2x 1 x x 2 x 1 x x 22 2 x 2x 1 2 x 1 2 x x 1 = + + = +

Nhn thy ( )1 c dng : ( ) ( ) ( )2f x 1 f x x 2 =

www.VNMATH.com


28/70


29/70

t2

t log x= . Lc : ( ) ( ) ( ) ( )21 x 1 .t 2x 5 .t 6 0 2 + + +

Lp ( ) ( ) ( )2 222x 5 24 x 1 4x 4x 1 2x 1 = + + = + = .

( ) ( )

1 2

2x 5 2x 1 2x 5 2x 1 3t 2 t

x 12 x 1 2 x 1

+ + + + = = = =

++ +.

Xt 1 23 2x 1

t t 2 x 1 x 1

= =+ +

x 1 0 1

2 +

1 2t t + 0 0 +

Nu1 2 1 2

10 x t t 0 t t ,

2< < < lc tp nghim ca ( )2 l :

( )( )

22 1

2 2 2

log x 2 at log x t3t log x t log x b

x 1

= = +

Do , khi1

0 x2

< th ( )a tha ( ), b khng tha nn tp nghim ( )2 l 10;2

( )3

Nu1 2 2 1

1x t t 0 t t ,

2> > < lc tp nghim ca ( )2 l

22 1

2 2 2

log x 2 x 4t log x t 3 1t log x t log x x 2

x 1 2

= = < +

Do , khi1

x2

> th tp nghim ca ( )2 l )1 ;2 4;2

+

( )4

T( ) ( )3 , 4 Tp nghim ca phng trnh l : ( )x 0;2 4; + .

Bi68.Bi68.Bi68.Bi68. i hc Nng Nghip I khi A nm 2001

Gii v bin lun bt phng trnh : ( )2 2a a aa a1

log log x log log x log 22+

Bi gii tham kho

iu kin : x 0> .

Cs a phi tha mn iu kin : 0 a 1< .

( ) a a a a a1 1 1

log . log x log log x log 22 2 2

+

a a a a a a

1 1 1log log log x log log x log 2

2 2 2 + +

a a a a1 3 1log log log x log 22 2 2

+

a a a

3 3log log x log 2

2 2

www.VNMATH.com


30/70

( ) a a alog log x log 2

Nu ( ) 2a0 a 1 : 0 log x 2 a x 1< < < < .

Nu ( ) 2aa 1 : log x 2 x a> .

Bi69.Bi69.Bi69.Bi69. Hc Vin Cng Ngh Bu Chnh Vin Thng nm 2001Tm tt c cc gi tr ca tham s a sao cho bt phng trnh sau c nghim ng x 0 :

( ) ( ) ( ) ( )x x

x 1a.2 2a 1 . 3 5 3 5 0+ + + + + <

Bi gii tham kho

( ) ( ) ( ) ( )x x

x2a 1 . 3 5 3 5 2a.2 0 + + + + <

( ) ( )x x

3 5 3 52a 1 . 2a 0 1

2 2

+ + + +


31/70

Gii phng trnh : ( ) ( ) ( ) ( )2 2

2x 33x 7log 9 12x 4x log 6x 23x 21 4

+++ + + + + =

Bi gii tham kho

iu kin :

( )( )( )

2

2

72 x1 3x 7 0

31 2x 3 0 3 31 x 1 x

29 12x 4x 0 2

2x 3 06x 23x 21 02x 3 3x 7 0

> + > + > > > + + >

+ > + + > + + >

.

( ) ( ) ( ) ( ) ( )( )2

3x 7 2x 3log 2x 3 log 2x 3 3x 7 4

+ + + + + + =

( ) ( ) ( ) ( ) ( ) ( ) 3x 7 2x 3 2x 32 log 2x 3 log 2x 3 log 3x 7 4+ + + + + + + + =

( ) ( ) ( ) ( ) ( )( )

( ) ( )

3x 7 3x 73x 7

2

3x 7

t log 2x 3 t log 2x 3 1t log 2x 3

1 12t 3t 1 02t 3 0 t log 2x 3

t 2

+ ++

+

= + = + = = + + =+ = = + =

( )

2

3xx 4 L

22x 3 3x 7 12x 3 0 x 2 x

42x 3 3x 719 12x 4x 3x 7

x4

= + = + + = = + = + + + = + =

.


x4

= .

Bi71.Bi71.Bi71.Bi71. i hc Thy Sn nm 1999Gii bt phng trnh : ( ) ( )x x2log 7.10 5.25 2x 1 > +

Bi gii tham kho

iu kin :

x

x x x x10

25

10 5 57.10 5.25 0 7.10 5.25 x log

25 7 7

> > > > >

xx

x

2

55 t 0t 0 2 52 1 1 x 025 22

5t 7t 2 0 t 15

= > = > < < < <


32/70

( )

2

22

4 17 14 17 1x 8x 10 4 17 x 5x 4 17x 4 17x 1

x 8x 1 4 17 x 1x 5x 4x 54 0x 1 1 x 1x 1

< < + > < > + > + + + + < + + +

.

Vy tp nghim ca bt phng trnh l : ( ( )x 4 17; 5 4 17;1 + .Bi73.Bi73.Bi73.Bi73. i hc Quc Gia H Ni khi D nm 1999

Gii bt phng trnh : ( ) ( )212

log x 3x 2 1 +

Bi gii tham kho

( ) ( ) ( )2 22 2log x 3x 2 1 log x 3x 2 1 + + 2

2

x 3x 2 0 x 1 x 2 0 x 1

0 x 3 2 x 3x 3x 2 2

+ > < >

( ) ( ) ( )x x x3m 1 .12 2 m 6 3 0+ + + < Bi gii tham kho

www.VNMATH.com


33/70

1/ Gii h phng trnh :

( ) ( )( )

x y

y x

3 3

4 32

log x y 1 log x y

+ = = +

iu kin : x y> .

( )( ) ( ) ( )( )

x y 5

y x 2

2 23 3 3

xt

yx y 51 54 4 ty x 2t 2log x y log x y log 3 x y x y 3

x y 3 0

+

= + = = + =

+ + = + = =

2 2

2 22 22 22 2

xx 1t y 2x

y y 2xy 2 x y 3 01x x 2yt t 2

2 x 2y2y x y 3 0x y 3 0

x y 3 0x y 3 0

= = = = = = = = = = = = = =

( )

2 2

y 2x x 2y y 1, x 2VN

y 1, x 23x 3 y 1

= = = = = = = =

.

Kt hp vi iu kin, nghim h phng trnh l ( ) ( ){ }S x; y 2;1= = .2/ Tm tt c cc gi tr ca m bt phng trnh sau c nghim ng x 0 : >

( ) ( ) ( )x x x3m 1 .12 2 m 6 3 0+ + + <

( ) ( ) ( ) ( )x x

3m 1 .4 2 m 2 1 0 1 + + + <

t xt 2 . Do x 0 t 1= > > . Lc : ( ) ( ) ( ) 21 3m 1 .t 2 m .t 1 0, t 1 + + + < >

( ) ( ) ( ) ( )2

2 2

2

t 2t 13t t m t 2t 1, t 1; m f t , t 1;

3t t

< + < = +

.

Xt hm s : ( )2

2

t 2t 1f t

3t t

=

trn khong ( )1;+ .

Ta c : ( )( )

( )2

22

7t 6t 1f ' t 0, t 1;

3t t

+ = > +

.

Bng bin thin

t 1 +

( )f ' t +

( )f t

1

3

2

Da vo bng bin thin, ta c: m 2< tha yu cu bi ton.

BBBBiiii77777777.... i hc Y Tp. H Ch Minh nm 1999

www.VNMATH.com


34/70

Gii phng trnh : ( )1999 1999sin x cos x 1+ = Bi gii tham kho

( ) 1999 1999 2 2 2 1997 2 19971 sin x cos x 0 sin x cos x sin x sin x cos x cos x 0 = + =

( ) ( ) ( ) 2 1997 2 1997sin x 1 sin x cos x 1 cos x 0 1 + =

Ta c : ( )( ) ( )

2 1997

2 1997

sin x 1 sin x 02

cos x 1 cos x 0

T( ) ( )( )( )

2 1997

2 1997

x k2sin x 1 sin x 0 sin x 0 cos x 01 , 2

cos x 1 sin x 1 x k2cos x 1 cos x 02

= = = = = = = + =

.

Bi78.Bi78.Bi78.Bi78. i hc Y Dc Tp. H Ch Minh nm 19991/ Gii bt phng trnh :

( )

( )( )

3a

a

log 35 x3, a 0, a 1

log 5 x

> >

.

2/ Xc nh m bt phng trnh : x x4 m.2 m 3 0 + + c nghim.

Bi gii tham kho

1/ Gii bt phng trnh :( )( )

( ) ( )3

a

a

log 35 x3 , a 0, a 1

log 5 x

> >

.

iu kin :3 3

335 x 0 x 35 x 355 x 0 x 5

> >

Do 3x 35 4 x 4 5 x 5 4 a 5 x 1< < > > = > nn :

( ) ( )33 21 35 x 5 x x 5x 6 0 2 x 3 > + < < < .

Kt hp vi tp xc nh, tp nghim ca bt phng trnh : ( )x 2; 3 .

2/ Xc nh m bt phng trnh : ( )x x4 m.2 m 3 0 + + c nghim.

tx

t 2 0= > . Lc : ( ) ( )2

t mt m 3 0, t 0; + + +

( ) ( ) ( ) ( ) { }2

2 t 3t 3 m t 1 , t 0; m f t , t 0; \ 1t 1

+ + + = +

.

Xt hm s ( )2t 3

f tt 1

+=

trn ( ) { }0; \ 1+

Ta c : ( )( )

( ) { }2

2

t 2t 3f ' t , t 0; \ 1

t 1

= +

. Cho ( )f ' t 0 t 1 t 3= = = .

Bng bin thin

t 1 0 1 3 +

( )f ' t + 0 0 +

www.VNMATH.com


35/70

( )f t 3 + +

6

Da vo bng bin thin, bt phng trnh c nghim : m 3 m 6< .

Bi79.Bi79.Bi79.Bi79. i hc Quc Gia Tp. H Ch Minh nm 1998Cho h phng trnh :

( ) ( )( )

2 2

m 3

9x 4y 5

log 3x 2y log 3x 2y 1

= + =

1/ Gii h( ) khi m 5= .

2/ Tm gi tr ln nht ca tham s m sao cho h( ) c nghim ( )x;y tha 3x 2y 5+ .Bi gii tham kho

1/ Khi m 5= th ( )

( ) ( )

( )2 2

5 3

9x 4y 51


= + =

iu kin:x 0, y 03x 2y 0

23x 2y 0 x y3

> > + > > >

.

( )( ) ( )

( )2 2

5 3

9x 4y 51


= + =

( )

( )( )

( ) ( )555

3x 2y 3x 2y 5

1 log 3x 2ylog 3x 2y 1log 3

+ =

+ =

( )( )( ) ( )

5 5 5 5

3x 2y 3x 2y 5

log 3. log 3x 2y log 3x 2y log 3

+ = + =

( )

5 5 5 5

53x 2y

3x 2y5

log 3. log log 3x 2y log 33x 2y

+ = =

( )( )( ) ( )

5 5 5 5 5

3x 2y 3x 2y 5

log 3. log 5 log 3x 2y log 3x 2y log 3

+ = =

( )( )( ) ( )

5 5 5 5 5

3x 2y 3x 2y 5

log 3 log 3. log 3x 2y log 3x 2y log 3

+ = =

( )( )( ) ( )

5 5

3x 2y 3x 2y 5

log 3 1 log 3x 2y 0

+ = =

( )( )( )

( )( )

5

3x 2y 3x 2y 5 3x 2y 5 x 13x 2y 3x 2y 5

3x 2y 1 y 13x 2y 1log 3x 2y 0

+ = + = = + = = = = =

.

So vi iu kin, nghim ca h phng trnh l ( ) ( ){ }S x; y 1;1= = .

www.VNMATH.com


36/70

2/ Tm gi tr ln nht ca tham s m sao cho h :( )

( ) ( ) ( )

2 2

m 3

9x 4y 5 2

log 3x 2y log 3x 2y 1 3

= + =

c nghim ( )x;y tha 3x 2y 5+ .

Ta c:( )( )3x 2y 3x 2y 5

3x 2y 13x 2y 5

+ = +

.

t5

t 3x 2y 3x 2yt

= + = .

( ) 3m 3 m 3 3 m3

1 log t5 53 log log t 1 log 3. log 1 log t log 3

t t 5log

t

+ = = + =

( ) 3m3 3

1 log tlog 3 4

log 5 log t

+ =

. t ( )3z log t, z 0 do t 3x 2y 1= = .

Lc : ( ) ( ) m3

z 14 log 3 f z , z 0z log 5+ = = +v 3z log 5 .

Xt hm s : ( )3

z 1f z

z log 5

+=

+trn ) { }30; \ log 5 + .

Ta c : ( )( )

) { }3 323

log 5 1f ' z 0, z 0; \ log 5

z log 5

+ = > + +

.

Bng bin thin

z 0 3log 5 +

( )f ' z 0 + +

( )f z + 1

5log 3

Da vo bng bin thin, phng trnh c nghim tha 3x 2y 5+ th

m 33

m 5 3 3

3 5

1 1 1log 3 1 log m 1log m m3

log 3 log 3 1 1 log m log 5 m 5log m log 3

.

Vy gi tr ln nht ca m l m 5= .

Bi80.Bi80.Bi80.Bi80. i hc Kinh T Tp. H Ch Minh khi A nm 1998Gii bt phng trnh :

( )( )

211

33

1 1

log x 1log 2x 3x 1>

+ +

Bi gii tham kho

www.VNMATH.com


37/70

iu kin :

2

2

1x x 12x 3x 1 0

2 11 x , x 032x 3x 1 1 2x 0, x

2 3x 1 0 x 1, xx 12x 1 1

x 0

< > + > < < + + > > > +

.

( ) ( ) ( ) ( )2 23 33 3

1 1 1 11log x 1 log x 1log 2x 3x 1 log 2x 3x 1

> < + + + +

Da vo iu kin, ta c bng xt du

x 1 0 1

2 1

3

2

( )3log x 1+ 0 + + +

23

log 2x 3x 1 + + 0 0 +

Da vo bng xt du, ta thy:

Nu 1 x 0 : < < VT VP> Bt phng trnh v nghim.

Nu1

0 x : VT VP2

< < < Bt phng trnh c tha.

Nu3

1 x : VT VP2

< < < Bt phng trnh c tha.

Nu3

x

2

> th

( ) ( ) ( ) ( )2 23 3 3 31

1 log 2x 3x 1 log x 1 log 2x 3x 1 log x 12

+ > + + > +

( ) ( ) ( )2 22 2

3 3log 2x 3x 1 log x 1 2x 3x 1 x 1 x 5 + > + + > + > .

Vy tp nghim ca bt phng trnh l ( )1 3

x 0; 1; 5;2 2

+ .

Bi81.Bi81.Bi81.Bi81. i hc Kin Trc H Ni nm 1998Gii bt phng trnh :

( )( )

21 22

1 1 0 1log 2x 1 log x 3x 2

+ > +

Bi gii tham kho

iu kin : 2

2

1x2x 1 0

22x 1 1 x 0

x 1 x 2x 3x 2 0

3 5x 3x 2 1 x 2

> > < > + > +

( )1 3 5

x ;1 2; \2 2

+

.

( )( )

( )

( )

22 2

2 222 2 2

log 2x 1 log x 3x 21 11 0 0

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

+ > >

+ +

www.VNMATH.com


38/70

( )( )

( )2 2

22 2

2x 1log

x 3x 2f x 0 2

log 2x 1 . log x 3x 2

+ = >

+.

Xt du ca : ( )2log 2x 1

( )21

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

2

< < < < < .

( )2log 2x 1 0 2x 1 1 x 1 > > > .

Xt du ca : 22

log x 3x 2 +

2 2

2

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

2 2

+ + < + < < < .

2 2

2

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

2 2

+ + > + > < > .

Xt du ca :2

2

2x 1logx 3x 2

+

2

2 2

2x 1 2x 1 1 1 13log 0 0 1 x

2 6x 3x 2 x 3x 2

+< < < < > >

+ +.

Bng xt du ca ( )f x :

x

1

2

1 13

6

+ 1 2

3 5

2

+ +

( )2log 2x 1 + +

22

log x 3x 2 + +

22

2x 1log

x 3x 2

+

+ + +

( )f x + +

Do , tp nghim ca ( )2 l1 13 3 5

x ;1 ;6 2

+ + + .

Bi82.Bi82.Bi82.Bi82. i hc Ngoi Thng khi D nm 1998Gii phng trnh : ( )2 3 2 3log x log x 1 log x. log x+ < +

Bi gii tham kho

iu kin : x 0> Tp xc nh : ( )D 0;= + .

www.VNMATH.com


39/70


40/70

Tha yu cu bi ton th { }a 1;2;3;4;5;6;7 .

2/ Gii phng trnh : ( ) ( ) ( )x x

x2 3 2 3 4 2 + + =

( ) ( )x x

2 3 2 32 1 3

4 4

+ + =

Nhn thy x 1= l mt nghim phng trnh ( )3 .

Xt hm s

x x

2 3 2 3y

4 4

+ = + trn .

Ta c :

x x

2 3 2 3 2 3 2 3y ' . ln . ln 0, x

4 4 4 4

+ + = + < V tri l hm

s gim.

Cn v phiy 1=

l hm hng. Do , phng trnh

( )3c nghim duy nht v nghim

l x 1= .

Bi85.Bi85.Bi85.Bi85. i hc K Thut Cng Ngh nm 19981/ Gii bt phng trnh : ( )x 3 x2 2 9 1+

2/ Gii phng trnh : ( )9 x4 log x log 3 3 2+ = Bi gii tham kho

1/ Gii bt phng trnh : ( )x 3 x2 2 9 1+

( )x x

x x2x

t 2 0 t 2 081 2 9 0 1 2 8 0 x 3

1 t 8t 9t 8 02

= > = > + +

.

2/ Gii phng trnh : ( )9 x4 log x log 3 3 2+ =

iu kin : 0 x 1< Tp xc nh : ( ) { }D 0; \ 1= + .

( )3

323

3 3

t log x 1 x 3t log x12 2 log x 3 0 1

2t 3t 1 0log x x 3t log x2

= = == + = + = == =

.

So vi tp xc nh, nghim ca phng trnh l x 3 x 3= = .

Bi86.Bi86.Bi86.Bi86. i hc Hng Hi nm 1998Gii phng trnh : ( )x 2 x 24 16 10.2 + =

Bi gii tham kho

iu kin : x 2 0 x 2 Tp xc nh : )D 2;= + .

( )x 2x 2 x 2

2x 2

2 8 x 2 3 x 11t 2 0 t 2 0

x 3t 8 t 2t 10t 16 0 x 2 12 2

= = == > = > == = + = ==

.

So vi tp xc nh, phng trnh c hai nghim : x 3 x 11= = .

Bi87.Bi87.Bi87.Bi87. i hc Dn Lp Vn Lang nm 1998

www.VNMATH.com


41/70


42/70

iu kin :x 1 0 x 1

x 3 0 x 3

+

.

Ta c : ( )( ) ( )( )

( )11

10 3 10 3 1 10 3 10 310 3

+ = = = ++

.

( ) ( ) ( )

x 3 x 1

x 1 x 3

10 3 10 3

+

+

+ < +

( )( )

2 3 x 5x 3 x 1 x 3 x 1 2x 100 0

x 1 x 3 x 1 x 3 x 1 x 3 1 x 5

< < + + < + < <

+ + + < lun c

nghim ng vi mi x.Bi gii tham kho

( ) ( ) ( ) ( ) ( ) ( )( )2 2

x x x x3 2 m 1 .3 2 m 1 1 0 3 1 2 m 1 3 1 0 + + > + + >

( )( ) ( )( ) ( )( ) x x x x x3 1 3 1 2 m 1 3 1 0 3 1 3 2m 3 0 + + + > + >

( ) x x3 2m 3 0 3 2m 3 > >

( ) ng x th ( ) cng ng x

( )

x 32m 3 0 do 3 0 m

2

< > .

Vy3

m2

tha yu cu bi ton.

Bi90.Bi90.Bi90.Bi90. i hc Dn Lp Ngoi Ng Tin Hc nm 1997Bit rng x 1= l 1 nghim ca bt phng trnh : ( ) ( ) ( )2 2m mlog 2x x 3 log 3x x+ + .Hy gii bt phng trnh ny.

Bi gii tham kho

iu kin : ( )2

2

x 0

2x x 3 0, x 1x ; 0 ;133x x 0 x

3

+ > >

.

V x 1= l mt nghim ca bt phng trnh ( ) ( )2 2m mlog 2x x 3 log 3x x+ + nn tac :

m mlog 6 log 2 0 m 1 < < .

Do ( ) 2 2 20 m 1 nn : 2x x 3 3x x x 2x 3 0 1 x 3< < + + .

Kt hp vi iu kin, tp nghim bt phng trnh l ( )1

x 1; 0 ; 33

.

Bi91.Bi91.Bi91.Bi91. i hc An Ninh i hc Cnh St khi A nm 1997Tm min xc nh ca hm s :

2

1 1y log

1 x 1 x

= + .

www.VNMATH.com


43/70

Bi gii tham kho

Hm s xc nh khi v ch khi :2

x 11 x 0

2x1 11 01

1 x 1 x 1 x

+

2

2

x 1 x 1x 2x 1

1 2 x 1 1 2 x 101 x

+ +

.

Vy min xc nh ca hm s l ) )D 1 2; 1 1 2; 1 = + .Bi92.Bi92.Bi92.Bi92. i hc Thy Sn nm 1997

Gii phng trnh : ( )2x 2 x2 3.2 1 0+ + = Bi gii tham kho

Tp xc nh : D = .

( ) ( )

( )

x

x2

x x2

t 2 0t 2 0 3 17

4. 2 3.2 1 0 t44.t 3t 1 0

3 17t L

4

= > = > + + = =

+ = =

( ) x 2 217 3 17 3

2 x log log 17 3 24 4

= = =

Vy nghim phng trnh l ( )2x log 17 3 2= .Bi93.Bi93.Bi93.Bi93. i hc Quc Gia Tp. H Ch Minh khi D nm 1997

Cho bt phng trnh : ( ) ( ) ( )2 25 51 log x 1 log mx 4x m+ + + + . Hy tm tt c cc gitr ca tham s m bt phng trnh c nghim ng vi mi x.

Bi gii tham kho

( ) ( ) ( ) ( )2 2

2 25 5 2

5 x 1 mx 4x mlog 5 x 1 log mx 4x m

mx 4x m 0

+ + + + + + + + >

( )( )

( ) ( )

( )( )

( )

f

2

2 2 2

2

2

5x 4x 5x m 15x 4x 5 m x 1 x 14xm x 1 4x g x m 2

x 1

+ = + + +

+ > =


44/70


45/70

( )( ) ( )22 2

1 x

x 0 ylog 1 2x 1 2x 2 1 4x 1 x 5x 2x 0 5

x2

+

= = + = = + + = =

.

So vi iu, nghim ca h l ( ) 5 5S x; y ;2 2

= = .

Bi95.Bi95.Bi95.Bi95. i hc Ngoi Thng khi D nm 1997Gii phng trnh : ( )x 1 x2 4 x 1+ =

Bi gii tham kho

Tp xc nh : D = .

t x2 t 0= > . Lc : ( ) 2t 1 2 x

t 2t x 1 0t 1 2 x

= + + =

=

.

Trng hp 1 : ( )xt 1 2 x 2 1 2 x 1= + = +

Ta c

( )( )( ) ( )

xf x 2 :

g x 1 2 x :

f 1 g 1

= = =

( )1 : c mt nghim duy nht l x 1= .

Trng hp 2 : ( )xt 1 2 x 2 1 2 x 2= =

iu kin :2 x 0

1 x 21 1 x 0

< >

.

Ta c : ( ( ) ( )( ) ( )

( )xf x 2 h 1 2

x 1;2 : 2 :h x 1 2 x h 2 1

= > = = < =

V nghim.

Vy phng trnh c nghim duy nht x 1= .

Bi96.Bi96.Bi96.Bi96. i hc Quc Gia Tp. H Ch Minh i hc Lut Tp. H Ch Minh nm 1996Cho phng trnh : ( ) ( ) ( )

tan x tan x

3 2 2 3 2 2 m+ + =

1/ Gii phng trnh khi m 6= .

2/ Xc nh m phng trnh ( ) c ng hai nghim trong khong ;2 2

.

Bi gii tham kho

1/ Khi m 6= th ( ) ( ) ( )( )

tan x

tan x tan x t 3 2 2 03 2 2 3 2 2 6

1t 6

t

= + > + + = + =

( ) ( )

( )

tan xtanx

tan x2

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

t 6t 1 0 t 3 2 2 3 2 2

= + = + = + > + = = + =

( ) tan x 1 x k , k4

= = + .

L hm tng.

L hm gim

www.VNMATH.com


46/70

2/ Tm m ( ) ( ) ( )tan x tan x

3 2 2 3 2 2 m+ + = c ng 2 nghim ;2 2

.

Ta c ( ) ( )tan x

2

t 3 2 2 0

t mt 1 0

= + > + =

.

Do x ; tan x2 2

.

Vy ta cn xc nh m phng trnh : 2t mt 1 0 + = c hai nghim phn bit dng.

( )

2m 4 0

P 1 0 m 2

S m 0

= > = > > = >

.

Vy khi m 2> th phng trnh ( ) c hai nghim phn bit x ;2 2

.

Bi97.Bi97.Bi97.Bi97. i hc Ngoi Thng nm 1996Tm nghim dng ca phng trnh : ( )2 2log 3 log 5x x x+ =

Bi gii tham kho

iu kin : x 0> (do nghim dng).

t t2

log x t x 2 0= = > .

( ) ( )t t

t t t 2 32 3 5 15 5

+ = + =

Nhn thy t 1= l mt nghim ca phng trnh ( ) .

Xt hm s ( )t t

2 3f t

5 5

= +

Ta c : ( )t t

2 2 3 3f ' t ln ln 0, t

5 5 5 5

= + < Hm s ( )f t nghch bin.

Mt khc y 1= l hm hng s( )//Ox .

Vy t 1= l nghim duy nht ca ( ) t 1x 2 2 2 = = = l nghim cn tm ca ( ) .

Bi98.Bi98.Bi98.Bi98. i hc Quc Gia Tp. H Ch Minh nm 1996Cho phng trnh : ( ) ( ) ( )

x x

2 3 2 3 m+ + =

1/ Gii ( ) khi m 4= .

2/ Tm m phng trnh ( ) c hai nghim.Bi gii tham kho

Tp xc nh : D = .

1/ Khi m 4= .

www.VNMATH.com


47/70

( ) ( ) ( )( ) ( )

xx

x x

2

t 2 3 0 t 2 3 02 3 2 3 4

1t 4t 1 0t 4

t

= + > = + > + + = + =+ =

( )

( )

( )

( )

x x

x x

t 2 3 0 2 3 2 3 x 1

x 1

t 2 3 2 3 2 3 2 3

= + > + = + = = = + = + =

.

Vy phng trnh c hai nghim x 1 x 1= = .

2/ Tm m phng trnh ( ) ( ) ( )x x

2 3 2 3 m+ + = c hai nghim.

( ) ( ) ( )( )

x

x x

2

t 2 3 0 t 02 3 2 3 m

1 t mt 1 0t m

t

= + > > + + = + = + =

2

m 2 m 2m 4 0 m 2m 0S m 0

< > = > >

>= >

.

Bi99.Bi99.Bi99.Bi99. i hc Quc Gia H Ni Hc Vin Ngn Hng nm 2000Gii phng trnh: ( ) ( ) ( )2 2

log x log x22 2 x. 2 2 1 x+ + = +

Bi gii tham kho

iu kin: x 1> Tp xc nh ( )D 1;= + .

tt 2 t

2log x t x 2 x 4= = = .

( ) ( ) ( )t t

t t2 2 2 2 2 1 4 + + = +

( ) ( ) ( )( )t tt

2 2 2 2 2 1 2 2 2 2 2

+ + = + +

t t t ta b 1 a b + = + vi( )

a 2 2

b 2 2 2

= + =

( ) ( ) t t t ta 1 b a b 0 + = ( ) ( )t t ta 1 b a 1 0 = ( )( )t ta 1 1 b 0 = t

2t

a 1t 0 log x 0 x 1

b 1

= = = =

=

.

Vy nghim phng trnh l { }S 1= .

Bi100.Bi100.Bi100.Bi100. i hc Quc Gia H Ni khi D nm 2000Gii phng trnh: ( )x x x8.3 3.2 24 6+ = +

Bi gii tham kho Tp xc nh: D = .

( ) ( ) ( )x x x x8.3 24 3.2 2 .3 0 + = ( ) ( )x x x8 3 3 2 3 3 0 =

www.VNMATH.com


48/70


49/70

Xt hm s ( )2

2

t t 2h t

t 2t

+=

+trn )1;6 .

Ta c: ( )( )

)2

2

3t 4t 4h ' t , t 1;6

t 2t

= +

. Cho ( )( )2

2

t 23t 4t 4

h ' t 0 2tt 2t

3

= = = = +

.

Bng bin thin

t 2

3 1 2 6 +

( )h ' t 0 0 +

( )h t

2

3

2

3

1

2

Da vo bng bin thin, ta c1

m2

tha yu cu bi ton.

Bi102.Bi102.Bi102.Bi102. i hc Bch Khoa H Ni khi D nm 2000Gii cc phng trnh: ( ) ( ) ( )

2 3

4 82log x 1 2 log 4 x log 4 x+ + = + +

Bi gii tham kho

iu kin:

( )

( )

2

3

x 1 0 x 1 0x 1

4 x 0 4 x 0 4 x 44 x 04 x 0

+ > + > > < + >

TX: ( ) { }D 4; 4 \ 1= .

( ) ( ) ( )2 2 2 2log x 1 log 4 log 4 x log 4 x + + = + +

( )( ) 22 2log 4 x 1 log 4 x 4 x 4 x 1 16 x + = + + =

( )

( )

( )

( )

( )( )

2

2

x 1

x 2 N4 x 1 16 x

x 6 Lx 2x 1 0

x 1 x 2 244 x 1 16 x

x 2 24 Nx 1 0

x 2 24 L

=+ = = =+ < = + = = + < = +

.

Vy nghim phng trnh l { }S 2 24; 2= .Bi103.Bi103.Bi103.Bi103. i hc SPhm H Ni khi A nm 2000

Tm m x 0;2

u tha mn bt phng trnh:

( ) ( )2 22 4log x 2x m 4 log x 2x m 5 + + + Bi gii tham kho

iu kin: 2x 2x m 0 + > .

www.VNMATH.com


50/70

t ( )24t log x 2x m 0= + .

( ) ( ) ( )2 2

4 4

2

t log x 2x m 0 t log x 2x m 0

5 t 1t 4t 5 0

= + = + +

( )( )

2 2

24 2 2

x 2x m 1 x 2x 1 m0 log x 2x m 1

x 2x m 4 x 2x 4 m

+ + +

.

Xt hm s ( ) 2f x x 2x, x 0;2 = .

Ta c: ( )f ' x 2x 2= . Cho ( )f ' x 0 x 1= = .Bng bin thin

x 0 1 2 +

( )f ' x 0 +

( )f x 0 0

1

Da vo bng bin thin v ( )1 m 1 m 2

2 m 44 m 0 m 4

.

Vy m 2; 4 tha yu cu bi ton.

Bi104.Bi104.Bi104.Bi104.i hc SPhm H Ni khi B, D nm 2000Gii bt phng trnh: ( )2x x x 4 x 43 8.3 9.9 0+ + + >

Bi gii tham kho

iu kin: x 4 0 x 4+ Tp xc nh: )D 4;= + .

Chia hai v cho x 4 2 x 49 3 0,+ += > ta c:

( ) ( )2 x x 4 x x 43 8.3 9 0

+ + >

x x 4x x 4x x 4

2

t 3 0t 3 0t 3 9 x x 4 2t 1

t 8t 9 0t 9

+ + +

= > = > = > + >< > >

( )

2

x 2 0

x 4 x 2 x 4 0 x 5

x 4 x 2

> + < + > + <

.

Kt hp tp xc nh, tp nghim bt phng trnh l

( )S 5;= + .

Bi105.Bi105.Bi105.Bi105. i hc SPhm Tp. H Ch Minh khi A, B nm 2000Gii bt phng trnh: ( ) ( ) ( )2 29 3log 3x 4x 2 1 log 3x 4x 2+ + + > + +

www.VNMATH.com


51/70

Bi gii tham kho

iu kin: 23x 4x 2 0, x+ + > Tp xc nh: D = .

t ( )23t log 3x 4x 2 ,= + + lc :

( )( )

2

t 1 0 t 1 01 1

t 1 t t t 1 0 t 21 12 2 t 0 t t 1

2 2

< + > > >

( )2 2

23 2 2

3x 4x 2 1 3x 4x 1 00 log 3x 4x 2 2

3x 4x 2 9 3x 4x 7 0

+ + + + + + < + + < + > + >

Tp xc nh : ( )D 0;= + .

t t7

log x t x 7= = . Lc : ( )t

t 23 3

t log 7 2 t log 7 2 = + = +

www.VNMATH.com


52/70

( )

t

23

tlog 7 2 ttt t 2

7 13 3 3 7 2 1 2. f t

3 3

+ = = + = + =

.

Xt hm s ( )t t

7 1f t 2.

3 3

= + trn .

Ta c: ( ) t t

7 7 1 1f ' t . ln 2. . ln 0, t

3 3 3 3

= + < Hm s ( )f t lun nghch bin

trn v c ( )2 2

7 1f 2 2. 1

3 3

= + = . V vy ( ) ( ) 2f t f 2 t 2 x 7 49= = = = .

So vi tp xc nh, nghim ca phng trnh l x 49= .

Bi108.Bi108.Bi108.Bi108. i hc Ngoi Thng khi A cs2 Tp. H Ch Minh nm 2000Gii bt phng trnh: ( ) ( ) ( )

x x

2 3log 2 1 log 4 2 2+ + + Bi gii tham kho

iu kin:x

x

2 1 0

4 2 0

+ > + >

ng x Tp xc nh : D = .

Xt hm s ( ) ( ) ( )x x2 3f x log 2 1 log 4 2= + + + trn .

Ta c: ( )

( ) ( )

x x

x x

2 ln 2 4 ln 4f ' x 0, x

2 1 ln 2 4 1 ln 3= + >

+ + Hm s ( )f x lun ng bin

trn v c ( ) 2 3f 0 log 2 lo 3 2= + = . Do : ( ) ( )x 0 f x f 0 x 0 .

Vy tp nghim ca bt phng trnh l (x ; 0 .

Bi109.Bi109.Bi109.Bi109. i hc Ngoi Thng khi D nm 2000Gii phng trnh : ( ) ( )2 23 3log x x 1 log x 2x x+ + =

Bi gii tham kho

iu kin:

2x x 1 0, x

x 0x 0

+ + > > >

Tp xc nh ( )D 0;= + .

( ) ( ) ( )2

223 3

x x 1 1log 2x x log x 1 1 x 1 1

x x

+ + = + + = .

Ta c: x 0 > thCauchy1 1 1

x 2 x. 2 x 1 3x x x

+ = + +

3 3 3

1 1log x 1 log 3 log x 1 1

x x

+ + + + .

Du " "= xy ra khi v ch khi21 x 1x 1x

x 1xx 0x 0x 0

= == = >> >

( )2 .

www.VNMATH.com


53/70

Mt khc: x 0 > th ( ) ( ) ( )2 2 2

x 1 0 x 1 0 1 x 1 1 . Du " "= xy

ra khi v ch khi x 1= ( )3 .

T( ) ( ) ( )1 , 2 , 3

( )

( ) ( )

2

3

3

3

22

1log x 1 1 x 1

x 1log x 1 11 xlog x 1 1 x 1

x 1 x 1 11 x 1 1

+ + = + + = + + = =

.

Vy nghim ca phng trnh l x 1= .

Bi110.Bi110.Bi110.Bi110. Hc Vin Quan H Quc T khi D nm 2000Gii phng trnh :

( ) ( ) ( ) ( ) ( )2 2 4 2 4 22 2 2 2log x x 1 log x x 1 log x x 1 log x x 1+ + + + = + + + + Bi gii tham kho

iu kin :

2

2 2

2

4 2 2

24 2

2

2

1 3x 0, x

2 4

x x 1 0 1 3x 0, x

x x 1 0 2 4

x x 1 0 1 3x 0, x

x x 1 0 2 4

1 3x 0, x

2 4

+ + > + + > + > + > + + > + + > + > + >

Tp xc nh D = .

( ) ( ) ( ) ( ) ( )2 2 4 2 4 22 2 2log x 1 x x 1 x log x x 1 log x x 1 + + + = + + + +

( ) ( ) ( )2

2 2 4 2 4 22 2 2

log x 1 x log x x 1 log x x 1 + = + + + +

( ) ( ) ( ) 4 2 4 2 4 22 2 2log x x 1 log x x 1 log x x 1 + + = + + + +

( ) 4 2 4 2 4 22log x x 1 0 x x 1 1 x x 0 x 0 x 1 + = + = = = = .

Vy tp nghim phng trnh l { }S 1; 0;1= .Bi111.Bi111.Bi111.Bi111. i hc Kinh T Quc Dn H Ni khi A nm 2000

Gii bt phng trnh : ( )2 25x 1

x 4x 3 1 log 8x 2x 6 1 05 x

+ + + +

Bi gii tham kho

iu kin :

2

2

x 4x 3 0 x 1 x 3x 1

2x 8x 6 0 1 x 3x 3

x x 005

+ = + = > >

.

Vi ( )x 1 : 0 0 := tha. Do , phng trnh c mt nghim x 1= .

www.VNMATH.com


54/70

Vi ( ) 5 5 33 1 3

x 3 : log log 0 :5 3 5

= + = khng tha do5 53

3log log 1 0

5> = .

Vy phng trnh c duy nht mt nghim l x 1= .

Bi112.Bi112.Bi112.Bi112. i hc Ti Chnh K Ton H Ni nm 2000Gii h phng trnh : ( )

8 8log y log x

4 4

x y 4

log x log y 1

+ = =

Bi gii tham kho

iu kin :x 0

y 0

> >

.

( )( )

8 8 8 8

8 8

log y log x log y log x

log y log 4y

4

x y 4 x y 4 x 4y

x xlog 1 4 4y y 4

y y

+ = + = = = = + =

8 8 8log 4y log 4y log 4y8 y

x 4y x 4y x 4y

log 4y log 2y y 4 y 2

= = = =+ = =

8 8 22 2

x 4y x 4y

1 2 1 1log 4 log y log y

log y 3 3 log y

= = + = + =

2

2

y 2 1x 4yyy 21

8log y 1 y x 8 18 xlog y 3x 4y 2

= = = = = = = == =

.

Vy tp nghim ca h l ( ) ( )1 1S x; y ; , 8;22 8

= = .

Bi113.Bi113.Bi113.Bi113. i hc M a Cht H Ni nm 2000Gii v bin lun theo tham s thc a h phng trnh :

( )( )

2a x y xy

x y a 1 1

2 .4 2 2+

+ + = =

Bi gii tham kho

T( )1 y 1 a x = . Thay vo ( )2 , ta c : ( )2 x 1 a x x 1 a xa2 .4 2

+ =

( ) ( )a

aa

22 2 22 1 x xa xa 1 x xa x 1 a 2 22 .4 2 2 2 2 1 x xa x 1 a

+ + + + = = + + =

( ) ( ) ( )222x 2 a 1 x a 1 0 3 + + = .

Lp ( ) ( ) ( )2 2 2

' a 1 2 a 1 a 1 0 = = .

Vi ( )a 1 : ' 0 3 : < v nghim h v nghim.

Vi ( ) 2a 1 : 3 2x 0 x 0 y 0= = = = .

www.VNMATH.com


55/70

Vya 1 :

a 1 :

=

Bi114.Bi114.Bi114.Bi114. i hc Lut i hc Xy Dng H Ni nm 2000

Gii bt phng trnh : ( )x

5 xlg

5 x 02 3x 1

+

< +

Bi gii tham kho

iu kin :x

5 x5 x 50

5 xx 1 x 3

2 3x 1 0

+ < +

Tp xc nh: ( ) { }D 5; 5 \ 1; 3= .

( ) x x x x

5 x 5 x 5 x 5 xlg 0 lg 0 1 1

5 x 5 x 5 x 5 x2 3x 1 0 2 3x 1 0 2 3x 1 2 3x 1

+ + + + > < > < + > +

2x 2x 0 x 5 x 0 x 50 05 x 5 x

x 1 x 3 1 x 3x 1 x 3 1 x 3

< < < >> < < <

.

Kt hp vi tp xc nh, tp nghim ca bt phng trnh l: ( ) ( )x 5; 0 1; 3 .

Bi115.Bi115.Bi115.Bi115. i hc Y H Ni nm 2000Gii cc phng trnh sau

1/( )

3x x

x3 x 1

1 122 6.2 1

22

+ = . 2/ ( ) ( )2 34 2lg x 1 lg x 1 25 + = .

Bi gii tham kho

1/ Gii phng trnh :( ) ( )

3x x

x3 x 1

1 122 6.2 1 1

22

+ =

Tp xc nh : D = .

( ) ( )( )

( )3

3x x x x

3x x 3 xx

8 12 8 21 2 6.2 1 0 2 6 2 1 0

2 2 22

+ = =

.

t x x2t 2 2= .

( ) ( )( ) ( )

( )( )

3 2 33 x x x x 3

x 2 3 3x x x

2 4 8 8t 2 3. 2 . 3.2 . 2 t 6t

2 2 2 2

= + = + .

( )( )

3x

x xx

xx

t 1t 6t 6t 1 2 1 Lx 122

t 2 2 2t 222

=+ = = = = ==

.

Vy nghim phng trnh l x 1= .2/ Gii phng trnh : ( ) ( ) ( )

2 34 2lg x 1 lg x 1 25 2 + =

H phng trnh v nghim.

H phng trnh c nghim .

www.VNMATH.com


56/70

iu kin :( )

( )

2

3

x 1 0 x 1 0x 1

x 1 0x 1 0

> > > >

Tp xc nh ( )D 1;= + .

( ) ( ) ( ) ( )4 2

4 22 2 lg x 1 3 lg x 1 25 0 16 lg x 1 9 lg x 1 25 0 + = + =

( )

( )

( ) ( )

22

2 2

25 x 1116t 9t 25 0 t 1 t Llg x 1 1

16 11t lg x 1 0 xt lg x 1 0 10

= + = = = = = > = = >

.

Kt hp tp xc nh, tp nghim ca phng trnh l11

S ;1110

=

.

Bi116.Bi116.Bi116.Bi116. i hc Y Thi Bnh nm 2000Gii bt phng trnh : ( )2 2xlog x log 8 4+

Bi gii tham kho

iu kin :x 0

10 x0 2x 1 2

> < <

Tp xc nh : ( ) 1D 0; \ 2 = +

.

( )( )

2

2 28

22

t 3t 11 1 0

log x 4 0 log x 4 0 t 1log 2x 1 t log x1 log x

3

+ + + =+

2

22

log x 13 13 3 13t 1 t

2 2 3 13 3 13log xt log x

2 2

< + < + =

3 13 3 13

2 21

x 2 x 22

+

< .

Kt hp vi tp xc nh, tp nghim ca h l3 13 3 13

2 21

x 0; 2 ; 22

+

.

Bi117.Bi117.Bi117.Bi117. i hc Y Hi Phng H chuyn ban nm 2000Tm x : ( ) ( ) ( )22 22 2 alog a x 5ax 3 5 x log 5 x 1+ + + = lun ng a .

Bi gii tham kho

iu kin cn : Nu h thc ng a th phi ng vi a 0= .

Lc : ( ) ( ) ( )2 2log 3 5 x log 5 x 1 3 5 x 5 x 1 + = + = .

( )( ) 5 x x 1 2 4 2 5 x x 1 4 x 5 x 1 + = + = = = .

iu kin :

Lc ( ) ( ) 222 2 ax 1 : log a 5a 5 log 5+= = . Hin nhin khng tha mn vi

( )5 5 5 5a 12 2

+< < .

www.VNMATH.com


57/70

Lc ( ) ( ) 222 2 ax 5 : log 25a 25a 3 log 3+= + = . Hin nhin khng tha mn vi

( )5 13 5 13a 210 10

+< < .

T( ) ( )1 , 2 khng c gi tr x tha yu cu bi ton.

Bi118.Bi118.Bi118.Bi118. i hc Ngoi NgH Ni H cha phn ban nm 20001/ Gii h phng trnh :

2 2 22x y z

xyz 64

= + =

vi ba s :y z x

log x, log y, log z theo th t

lp thnh cp s nhn.

2/ Cho phng trnh : ( ) ( )x xm 3 16 2m 1 4 m 1 0+ + + + = . Tm m phng trnh chai nghim tri du.

Bi gii tham kho

1/ Gii h phng trnh : ( )2 2 22x y z

xyz 64

= +

=

vi ba s :y z x

log x, log y, log z theo th t

lp thnh cp s nhn.

iu kin: 1 x, y, z 0 > .

Doy z x

log x, log y, log z theo th t lp thnh cp s nhn nn ta c:

2 2 2 3z y x z y z z

z

1log y log x. log z log y log z log y log y 1 z y

log y= = = = = .

( )

2 2

2 3

2x 2y x y

xy 64 y 64 x y z 4

1 x, y, z 0 1 x, y, z 0

= = = = = = =

> >

.

Vy nghim ca h l ( ) ( )x; y; z 4; 4; 4= .

2/ Cho phng trnh : ( ) ( ) ( )x xm 3 16 2m 1 4 m 1 0+ + + + = . Tm m phng trnh

( ) c hai nghim tri du. Tp xc nh : D = .

t xt 4 0= > . Khi : ( ) ( ) ( ) ( ) ( )2f t m 3 t 2m 1 t m 1 0 = + + + + = .

Gi 1 2x , x l hai nghim ca ( ) v 1 2t , t l hai nghim ca ( )

( ) c hai nghim tri du 1 2x 0 x < < 1 2x x

0 4 1 4 < < < 1 2

0 t 1 t < < <

( ) ( )( )( ) ( )

( )( )( )( )

m 3 .f 1 0 m 3 4m 3 0 31 m

m 3 m 1 .f 0 0 m 3 m 1 0 4

+ < + + + + >

.

Vy3

m 1;4

tha yu cu bi ton.

Bi119.Bi119.Bi119.Bi119. i hc Nng nm 2000Gii bt phng trnh : ( )x1 log 2000 2+ <

Bi gii tham kho

( ) ( )x x2 1 log 2000 2 3 log 2000 1 < + < < <

www.VNMATH.com


58/70

Trng hp 1 : ( )3

x 1

x 1 : x 200012000 x

x

>> > < .

( ) ( )x x 3 x x

2 x

8

log 9 2 3 x 9 2 2 2 9 02

= = + =

2 x

xx x

t 1 t 8t 9t 8 0 2 1 x 0

x 3t 2 0t 2 0 2 8

= = + = = = == >= > =

.

So vi iu kin, nghim ca phng trnh l: x 0= v x 3= .

Bi121.Bi121.Bi121.Bi121. i hc Hu khi D, R, R H chuyn ban nm 2000Gii phng trnh : ( ) ( ) ( )22 1

2

log x 1 log x 1 =

Bi gii tham kho

iu kin :2 x 1 x 1x 1 0

x 1x 1x 1 0

< > > > > >

Tp xc nh : ( )D 1;= + .

( ) ( ) ( ) ( )( ) ( )( )2 2 22 2 2log x 1 log x 1 0 log x 1 x 1 0 x 1 x 1 1 + = = =

( ) 2 1 5 1 5x x x 1 0 x 0 x x2 2

+ = = = = .

So vi tp xc nh, nghim ca phng trnh l :1 5

x 2

+= .

Bi122.Bi122.Bi122.Bi122. i hc SPhm Vinh khi D, G, M nm 2000Gii phng trnh : ( ) ( ) ( ) ( )x 1 x5 5 5x 1 log 3 log 3 3 log 11.3 9+ + + =

Bi gii tham kho

iu kin : x 1 x3 3 0 11.3 9 0+ + > > .

( ) ( ) ( )x 1 x 1 x5 5 5log 3 log 3 3 log 11.3 9 + + + =

( ) ( ) x 1 x 1 x 2x x x5 5log 3 . 3 3 log 11.3 9 3 3 11.3 9 + + = + =

( )x

2x x

x

3 1 x 03 10.3 9 0

x 23 9

= = + = ==

.

www.VNMATH.com


59/70

So vi iu kin, nghim ca phng trnh l: x 0, x 2= = .

Bi123.Bi123.Bi123.Bi123. i hc Cng on nm 2000Gii phng trnh : ( )3 32 2

4log x log x

3+ =

Bi gii tham kho

iu kin :

3 x 0

x 0x 0

> > >

Tp xc nh : ( )D 0;= + .

( )33 3

2 223

2 2 3

t log x t log x log x t1 4log x log x 0 1 4 t 13 3 t t 0

3 3

= = = + = =+ =

x 2 =

So vi tp xc nh, nghim ca phng trnh l x 2= .

Bi124.Bi124.Bi124.Bi124. i hc Thy Li H Ni H cha phn ban nm 2000

Gii h phng trnh : ( )2 2 23 3 3

3xx log 3 log y y log

22y

x log 12 log x y log3

+ = + + = +

Bi gii tham kho

iu kin : x 0, y 0> > .

( )( )

( )

x yx y

2 22 2 2 2

x y x y3 3 3 3 3 2

3x3x log 3 .y log 2 .log 3 log y log 2 log 222y 2y

log 12 log x log 3 log log 12 .x log 3 .3 3

= + = + + = + =

( )

( )

( )( )

1

x y 2 x yx y 2x y

y xy x

3x3 .y 2 . 1 3 3 2 32 . . 36 6 6 6 y 2x

2y 2 23 123 . 12 .x 23

= = = = = =

.

Thay y 2x= vo ( )1 , ta c : ( )x 1

x 2x x 1 x 13x 31 3 .2x 2 . 3 4 12 4

= = =

x 1 0 x 1 y 2 = = = .

Vy nghim ca h l ( ) ( ){ }S x; y 1;2= = .

Bi125.Bi125.Bi125.Bi125.

Documents

200 BAI Mu Logarit