Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh
hÖ ph-¬ng tr×nh, hÖ bÊt ph-¬ng tr×nh mò
VÊn ®Ò 1.
L o¹i 1:
1.
2.
3.
4.
5.
6.
4 x = 8 2x – 1,
5 2x = 625
16 -x = 8 2(1 – x) ,
2
2 x -3 x + 2 = 4
6 3-x = 216
2
3 4 - 2 x = 9 5-3 x - x
1
15. 5
16.
17.
18.
19.
L o¹i 2:
1.
2.
L o¹i 3:
1.
2.
3.
4.
5.
6.
7.
8.
2
x +5
= 625
32. 2 x
3 < 27.9
2
2 2 x -5 x -1 = 0,125
x2
(
)
2 +1
( 5 + 2)
x +1
£
(
2
)
2 -1
-x
³ ( 5 - 2)
x -1
x +1
9
x+
7
2 2 x +5 - 3 2 = 3 2 - 4 x + 4
1
1
3.4 x + .9 x + 2 = 6.4 x + 2 - .9 x +1
3
2
7.3 x +1 + 5 x +3 £ 3 x + 4 + 5 x + 2
2
-4
+ x -2
x
æ1ö
43. æç 3 3 3 ö÷ = ç ÷
è
ø
è 81 ø
2
1 x - x -1
44. 3 x - 2 x ³ ( )
3
(
)
é
46. ê2 2
ë
= 81 3
= 16 2
(
é
47. ê 2
ë
x +7
x -3
2 x +3
(BKHN’98)
2 x 3 4 x x 0,125 = 3 0,25
45.
x +5
x -5
33. 32
= (0,25).(128)
2x
34. 5 = (0,04) 2 x -3
35. 5 2.5 4...5 2 x = 0,04 -28
3.2 x + 1 + 5.2 x – 2 x + 2 = 21
3 x – 1 + 3 x + 3 x + 1 = 9477
5 x + 1 – 5 x = 2 x + 1 + 2 x + 3,
2 x – 1 – 3 x = 3 x – 1 – 2 x + 2,
5 x + 5 x +1 + 5 x + 2 = 7 x + 7 x +1 - 7 x + 2
x+
2
-6 x - 2, 5
x +5
x -7
5 x - 2.2 x .3 x -1 = 12
64 2 x = 0,125
6 x -6
x +1
x 2 -6 x + 5
2
2
x
31. 3
3 x -7
1
-
= 10000
1 x - x -1
41. 3 x - 2 x ³ ( )
(LuËt’96)
3
2
42. (2 - 3 ) x +1 > (2 - 3 ) 3 x +1
40. 10 x
x +17
= (0,125).8
2
2
2
= 2 2 (x +1) + 2 2(x + 2 ) - 2 x +3 + 1
+4
1
28. 5 x .5 2 = 225
29. 5 2 x +1 - 3.5 2 x -1 = 550
30. 16
2
38. 16 x + 2 x -2 = (0,25).2 x
39. 3 x +1 = 18 2 x .2 -2 x .3 x + 2
26. 32 x -7 = 0,25.128 x -3
5
9 2
5
27. ( ) x +1 .( ) x + x -1 = ( ) 9
3
25
3
x +10
x -10
2x
37. 2 x + 2 - |2 x + 1 - 1| = 2 x + 1 + 1
2
22. 0.125.4 2 x -3 < ( ) - x
8
1
23. 2 cos 2 x =0
2.2 cos 2 x
24. 10 x+10x-1=0,11
3 3
25. ( 3 ) tg 2 x - tg 2 x = 0
3
4
x
36.
3 x +1
20. 3
³ 3 2 x +1
21. (0,4) x -1 = (6,25) 6 x -5
æ1öx æ1ö
7. ç ÷ > ç ÷
è2ø
è2ø
1
1 x
x -1
8. 2 > ( )
16
1
9. 3 x -1 =
729
10. 2 3x = (512) -3x
2
1
11. 3 x -4 x +1 =
9
2x
3
12. 128 = 4
13. 5 |4x - 6 | = 25 3x – 4
14. 3 |3x - 4 | = 9 2x – 2
x
§-a vÒ cïng c¬ sè
x +3 2
)
1
ù
x
ú
û
ù
x +1
ú
û
1
x +5 5
2
x -1
=4
1
x
1
= .4
2
x
48. ( 1 ) 2- x + 3 x-3 = 99 + ( 1 ) 4- x
3
( )
4. ( 10 + 3)
3. 2 - 3
9.
x 2 +1
( )
= ( 10 - 3)
> 2+ 3
x -3
x -1
9x - 2
x+
10. 4 - x - 3
x+
1
2
9
3
2
-x-
=2
1
2
x+
1
= 32
3 x +1
1
2
-x
x +1
x +3
(GTVT ’98)
- 3 2 x -1
- 2 -2 x -1
x-
1
- 9 x = 3 2 x -2 - 5 2
x +1
x +3
1
1
1
1
12. ( ) 2 x +3 - ( ) 2 > ( ) 2 - ( ) 2 x +1
2
3
3
2
13. 4 x + 2 – 10.3 x = 2.3 x + 3 – 11.2 2x
14. 3 x + 3 x +1 + 3 x + 2 £ 5 x -1 + 5 x + 5 x +1
11. 5
T rang 1
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
L o¹i 5:
1. x 2 - (2 x - 3) x + 2(1 - 2 x ) = 0
4. (4 x - 2)(3 2- x + 3 - 2 x) ³ 0
5. (3 x - 2 x - 1)(2 x - 1) < 0
6. x2.2 x + 1 + 2 |x - 3 | + 2 = x2 .2 |x - 3| + 4 + 2 x – 1
2. 4 x 2 + x.2 x +1 + 3.2 x > x 2 .2 x + 8 x + 12 (D-îc’97)
3. (3 x - 1)(31- x - 3 x + 1) > 0
L o¹i 6:
1. 6 2x + 3 = 2 x + 7.3 3x – 1,
3. 3 x +3.7 x +3 = 3 2 x. 7 2 x
x – 1 2x – 2
9 –x
2. 3 .2
= 12 ,
4. 6 2 x + 3 ³ 2 x + 7.33 x -1
2
2
2
5. 3 2 x +3.5 2 x +3 = 5 5 x .35 x
Gi¶i bpt víi a>0, a ¹ 1, x Î N *
1 + a 2 + ... + a x -1 + a x = (1 + a )(1 + a 2 )(1 + a 4 )(1 + a 8 )
VÊn ®Ò 2.
D ¹ng 1: § Æt Èn phô lu«n.
L o¹i 1:
1. 4 x + 2 x - 6 = 0
2. 4 x + 1 + 2 x + 4 = 2 x + 2 + 16
3. 9 x - 25.3 x + 7 = 0
4. 25 x - 23.5 x - 5 = 0
5. 25 x -6.5 x +1 + 5 3 = 0
1
6. 3.5 2 x -1 - 2.5 x -1 =
5
7. 13 2 x - 6.13 x + 5 = 0
3
8.
= 4 x -4 - 7
3- x
2
9. 3 2 ( x +1) - 82.3 x + 9 = 0
10. 4 x +1 + 2 x + 4 = 2 x + 2 + 16
11. 3 x + 2 + 9 x +1 = 4
§Æt Èn phô
16. 8 x - 3.4 x - 3.2 x +1 + 8 = 0
17. 4 2 x + 2 3 x +1 + 2 x + 2 - 16 = 0
2
18. 5 2 x -3 = x -1 + 15
5
2 x+6
19. 2
+ 2 x + 7 - 17 > 0
(NNHN’98)
20.
21.
22.
23.
24.
5 x -1 + 5 x -3 = 16
51+ x + 51- x = 16
3 2+ x + 3 2- x = 30
4 x + 2 3- 4 x = 6
1- x
3
x
-3
25. 5
x
- 51-
x
+4=0
=4
x
10 + 4 2
32. x -2 =
4
2
1
1
33. ( ) 3 x - ( ) x -1 - 128 ³ 0
4
8
x -2 x - x
34. 9
- 7.3 x - 2 x - x -1 = 2
9
2
2
x -1
x -1
35. 3.2 x +1 - 8.2 2 + 4 = 0
36. 5.2 3|x - 1| - 3.2 5 – 3x + 7 = 0.
3 x +3
x
2
x
37. 8 - 2
+ 12 = 0
4
x +4 x
38. 8.3
+ 9 x +1 = 9
39. 13 2 x - 6.13 x + 5 ³ 0
x
40. 9 x -3 + 3 < 28.3 -1+ x -3
26. 101+ x - 101- = 99
(PVBChÝ’98)
2
2
1+ x 2
1- x 2
41. 4 sin px + 3.4 cos px £ 8
27.
5
5
=
24
x 2 -1
x 2 +1
12. 9
-3
-6 = 0
æ πö
sin 2 ç x - ÷
1
x 2 -1
x 2 -3
æ pö
è 4ø
28. ( ) x -2 = 2 5- x + 9
tg ç x - ÷
13. 9
- 36.3
+3= 0
è 4ø
cos 2 x
4
42. 2
- 2.0,25
³1
14.
32 x
x
x -10
5
10
( 3) + ( 3)
- 84 = 0 29.
= 2(0.3) x + 3
2
2
100 x
43. 2 sin x + 4.2 cos x = 6
2
2
44. cotg2 x = tg2 x + 2tg2 X + 1
30. 4 x + 2 3- 4 x < 6
15. 4 x + x -2 - 5.2 x -1+ x -2 = 6
1
2
2
31. ( ) x -3 = 6 5- 2 x - 12
45. 81sin x + 81cos x = 30
6
L o¹i 2: § Æt Èn phô nh-ng vÉn cßn Èn x
1. 9 x + 2( x - 2).3 x + 2 x - 5 = 0 (§N’97)
6. 9 - x - ( x + 2).3 - x - 2( x + 4) = 0
2
2
2. 25 x - 2(3 - x).5 x + 2 x - 7 = 0 (TC’97)
7. 9 x + ( x 2 - 3).3 x + 2(1 - x 2 ) = 0
3. 3.16 x -2 - (3 x - 10).4 x - 2 + 3 - x = 0
8. 3 2 x + 2( x - 2).3 x + 2 x - 9 = 0
4. 3.4 x + (3 x - 10).2 x + 3 - x = 0
9. 3 2 x -3 - (3 x - 10).3 x -2 + 3 - x = 0
5. 8 - x.2 x + 2 3- x - x = 0
2
x2
2
2
T rang 2
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
1
x
11. 3.25
1
x
10. 4 + 2 x.2 - 6 x = 9
D ¹ng 2: C hia xong ®Æt
V Ý dô. Gi¶i ph-¬ng tr×nh:
27 x + 12 x + = 2.8 x
X -2
+ (3x - 10)5 x - 5 + 3 - x = 0
(1)
Gi¶i:
æ3ö
ç ÷
è2ø
3x
x
x
æ3ö
+ç ÷ = 2
è2ø
æ3ö
(2). § Æt ç ÷ = t
è2ø
(* ). K hi ®ã ph-¬ng tr×nh (2): t3 + t –2 = 0 , t > 0 .
x
æ3ö
t = 1 Þ ç ÷ = 1 suy ra x = log 3 1 = 0 . V Ëy ph-¬ng tr×nh ®· cho cã mét nghiÖm: x = 0 .
è2ø
2
Bµi tËp t-¬ng tù
1. 8 x + 18 x = 2.27 x
2. 6.9 x - 13.6 x + 6.4 x = 0
3. 4 x = 2.14x + 3.49 x.
4.
5.
6.
7.
8.
9.
10.
8x
3
-1
+ 18 x
3
-1
= 2.27 x
3
-1
3.4 x - 2.6 x = 9 x
2.4 x +1 + 6 x +1 = 9 x +1
3.16 x + 2.81x = 5.36 x
25 x + 10 x = 2 2 x +1 (HVNH’98)
125 x + 50 x = 2 3 x +1 (QGHN’98)
42 2 x - 6 x = 18.3 2 x
2
2
1
1
2
1
11. 49 x - 35 x = 25 x
12. 3 2 x + 4 + 45.6 x - 9.2 2 x + 2 = 0
1
x
1
x
1
x
13. 6.9 - 13.6 + 6.4 = 0 (TS’97)
D ¹ng 3: A x.Bx = 1.
14.
15.
16.
17.
18.
x
2
4.3 - 9.2 = 5.6
2 x -1 (2 x + 3 x -1 ) = 9 x -1
x
x
2.4
x+4 x
20. 8 .3
+ 9 1+
4
x
>9
x
21. 3 x +1 - 2 2 x +1 - 12 2 < 0 (HVCNBCVT’98)
cos x -sin x
æ1ö
2 sin x - 2 cos x +1
- 7.ç ÷
+ 5 2 sin x -2 cos x +1 = 0
22. 2
è 10 ø
æ1ö
è6ø
cos 2 x - 2 in 2 x - log6 14
23. 2 2 sin 2 x- cos 2 x+3 - ç ÷
(
) + (4 -
+ 3 2 sin 2 x-2 cos 2 x+1 = 0
)
2. (5 + 2 6 ) x + (5 - 2 6 ) x = 10
11. æç 3 3 - 8 ö÷ + æç 3 3 + 8 ö÷ = 6
è
ø è
ø
x
+
(5 + 2 6 )
x
4. (2 - 3 ) x + (2 + 3 ) x = 14 (NT’97)
6. ( 2 - 3 ) x + ( 2 + 3 ) x = 4
7. (3 + 5 ) + 7(3 - 5 ) = 2
x
(NN§N’95)
x
15
x
= 62
x
12. ( 7 + 48 ) x + ( 7 - 48 ) x = 14
13. (2 - 3 ) x
5. (2 - 3 ) x + (2 + 3 ) x = 4
x
x
x
= 10
= 3.9
-1
x
x
10. 4 + 15
(5 - 2 6 )
-6
-1
x
2.81x - 7.36 x + 5.16 x = 0
2.14 x + 3.49 x - 4 x ³ 0 (GT’96)
4 x - 2.6 x = 3.9 x (§HVH’98)
2
19. 4 lg( 20 x ) - 6 lg x = 2.3lg(100 x ) (BKHN’99)
1. (5 + 24 ) x + (5 - 24 ) x = 10
3.
-1
x
2
- 2 x -1
14. æç 2 - 3 ö÷
è
ø
+ (2 + 3 ) x
x 2 - 2 x -1
2
- 2 x +1
+ æç 2 + 3 ö÷
è
ø
=
2
2- 3
x 2 - 2 x +1
£
4
2- 3
8. (3 + 2 2 ) tgx + (3 - 2 2 ) tgx = 6
15. (5 - 21) x + 7(5 + 21) x = 2 x + 3 (QGHN’97)
9. (7 + 4 3 ) sin x + (7 - 4 3 ) sin x = 4
16. (2 + 3 ) x + (7 + 4 3 ).(2 - 3 ) x = 4(2 + 3 ) (NN’98)
T rang 3
17. æç 7 + 4 3 ö÷
è
ø
cos x
+ æç 7 - 4 3 ö÷
è
ø
cos x
(
19. 2 3 + 11
=4
)
(
+ 2 3 - 11
)
2 x -1
=4 3
20.
(L uËt HN’98)
18.
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
2 x -1
( 11 - 6 ) + ( 11 + 6 ) = ( 5 )
x
x
x
D ¹ng 4:
1. 4 x + 4 -x + 2 x + 2 -x = 10
4. 8 x + 1 + 8.(0,5) 3x + 3.2 x + 3 = 125 – 24.(0,5) x.
2. 3 1 – x – 3 1 + x + 9 x + 9 -x = 6
8
æ
3. ç 2 3 x - 3 x
2
è
5. 5 3x + 9.5 x + 27.(5 -3x + 5-x) = 64
1 ö
ö
æ x
÷ - 6.ç 2 - x -1 ÷ = 1
2 ø
ø
è
VÊn ®Ò 3.
D ¹ng 1:
V Ý dô. Gi¶i ph-¬ng tr×nh:
Sö dông tÝnh ®ång biÕn nghÞch biÕn
4 x + 3 x = 5 x (1)
Gi¶i:
C¸ch 1: Ta nhËn thÊy x = 2 lµ métnghiÖm cña PT (1), ta sÏ chøng minh nghiÖm ®ã lµ duy nhÊt.
x
Chia 2 vÕ cña ph-¬ng tr×nh cho 5 , ta ®-îc:
x
æ4ö
æ4ö
ç ÷ <ç ÷
è5ø
è5ø
+ V íi x > 2, ta cã:
2
x
;
x
æ 4ö æ3ö
ç ÷ +ç ÷ =1
è 5ø è5ø
x
2
(1')
x
x
2
2
æ3ö
æ 3ö
æ 4ö æ 3ö
æ 4ö æ3ö
ç ÷ < ç ÷ . Suy ra: ç ÷ + ç ÷ < ç ÷ + ç ÷ = 1
è5ø
è5ø
è5ø è5ø
è 5ø è5ø
§ iÒu nµy chøng tá (1') (hay(1)) kh«ng cã nghiÖm x > 2.
x
æ4ö
æ4ö
+ V íi x < 2, ta cã: ç ÷ > ç ÷
è5ø
è5ø
2
x
;
2
x
x
2
2
æ3ö
æ 3ö
æ 4ö æ3ö
æ 4ö æ 3ö
ç ÷ > ç ÷ . Suy ra: ç ÷ + ç ÷ > ç ÷ + ç ÷ = 1
è5ø
è5ø
è5ø è5ø
è5ø è5ø
§ iÒu nµy chøng tá (1') (hay(1)) kh«ng cã nghiÖm x < 2.
V Ëy ph-¬ng tr×nh ®· cho cã duy nhÊt mét nghiÖm x = 2 .
C¸ch 2: Ta thÊy x = 2 lµ nghiÖm cña ph-¬ng tr×nh (1 ’), ta chøng minh nghiÖm ®ã lµ duy n hÊt.
x
x
æ 4ö æ3ö
f ( x) = ç ÷ + ç ÷ . H µm sè f(x) x¸c ®Þnh víi mäi x Î R.
è5ø è5ø
§ Æt:
x
x
Ta cã:
4 æ 3ö
3
æ4ö
f ' ( x) = ç ÷ . ln + ç ÷ . ln < 0 , " x.
5 è5ø
5
è5ø
D o ®ã:
+ N Õu x > 2 th× f(x) > f(2) = 1
N h- vËy hµm sè f(x) ®ång biÕn " x Î R.
+ N Õu x < 2 th× f(x) < f(2) = 1 . V Ëy ph-¬ng tr×nh ®· cho cã nghiÖm duy nhÊtx = 2 .
B µi tËp t-¬ng tù:
x
2
1. 1 + 3 = 2
x
2. 2 x + 3 x = 5 x
3. 4 x = 3 x + 1
4. 7
x
2
+ 3x = 4 x
T rang 4
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
5. 8
x
+1 = 3
2
x
x
x
9. 2
x
x
2
+1 = 4x
+1 = 3
2
x
x
11. 4 + 9 = 25
12. 8 x + 18x = 2.27 x.
13. 2 x + 3 x + 1 > 6 x
14. 2.2 x + 3.3 x + 1 < 6 x
15. 3 x + 1 + 100 = 7 x –1
16. 4.3 x - 41- x = 11
17. 2 x + 3 x + 5 x = 38
2
6. 1 + 7 3 = 2 x
7. 3 x – 4 = 5 x/2
8. 15
x
2
2 + 3x 5
3 x + 4x + 8 x < 15 x
£
x
x
7
3 +4
19. 4 x + 9 x + 16x = 81 x
6 x + 8 x = 10 x
20. (6 - 4 2 )x + (17 - 12 2 )x + (34 - 24 2 )x ³ 1
x
18.
21. 4 x + 9 x +
( 32 )
x
= 13 x
22. ( 4 + 15 ) x + ( 4 - 15 ) x = (2 2 )
x
10. 4 x + 3 x = 5 5 x
x
x
23. æç 2 + 3 ö÷ + æç 2 - 3 ö÷ = 2 x
è
ø
è
ø
1. Gi¶i ph-¬ng tr×nh:
1.
3. x log 2 9 = x 2 .3 log 2 x - x log 2 3
x 2 + 3log 2 x = x log 2 5
4.
2. 4 log3 x = 2 + x log3 2
2. T×m c¸c gi¸ trÞ cña tham sè m ®Ó bÊtph-¬ng tr×nh sau lu«n cã nghiÖm: 2 sin x + 3 cos 2 x ³ m.3sin x
D ¹ng 2:
x
1. 4 x + 3 x - 7 = 0
9. 3 x + 5 x = -6 x + 2
æ1ö
6.
=
x
+
6
ç ÷
2. 3 x + x - 4 = 0
10. 3 x + 2 x = -3x + 2
è2ø
x
3. 5 + 4 x - 7 = 0
7. 2 x + 2 x - 14 = 0
4. 2 x = 3 – x
8. 7 x + 6 x = -11x + 2
5. 5 x + 2x – 7 = 0
D ¹ng 3: f(x) ®ång biÕn (nghÞch biÕn), f(x 1 ) = f(x2) Û x1 = x2.
2
1- x 2
1. 2 cos x - 2 sin x + cos 2 x = 0
2
2
2. e cos x - e sin x = cos 2 x
2
2
3. 2
x 2 - 3 x +1
4. 2 x
2
- 3 x +1
5. 2
x2
1- 2 x
-2
x2
6. 2 2 x -1 + 3 2 x
- 2 x -2 + x 2 - 3x - x + 3 = 0
7. 7
- 2 x -2 + x 2 - 4 x + 3 = 0
VÊn ®Ò 4.
log 5 ( x -1)
2
1 1
2 x
+ 5 2 x + 1 = 2 x + 3 x + 1 + 5 2 x +1
=
- 5 log 7 ( x +1) = 2
NhËn xÐt ®¸nh gi¸
Gi¶i c¸c ph-¬ng tr×nh sau:
1. 2 |x| = sinx2,
2.
4
2
x
+ 3cos
7. 2 x + ( 3 ) x = 21- x
5. 3 x + 2 x = -3 x 2 + 2
8. 2 x + 3 x + 7 x + 8 x = 41- x
6. 2 x + 3 x + 4 x = 3
9. 2 x + 4.10 x = 7 - 3 x
2
16 - x 2 = 2 x + 2 - x
3. 3sin
4. 5 x = cos 3 x
2
x
=4
2
VÊn ®Ò 5.
V Ý dô. Gi¶i ph-¬ng tr×nh:
Gi¶i:
(
)
2
2
2
2
2
2
2
2
2
2
2
2
2
Ph-¬ng ph¸p l«garÝt ho¸
3 x .2 x = 1
2
éx = 0
log 3 3 x.2 x = log 3 1 Û x + x 2 log 3 2 = 0 Û x(1 + x log 3 2 ) = 0 Û ê
2
ê x = - 1 = - log 2 3
log 3 2
ëê
T rang 5
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
1. 2 = 3
x
x +1
1
>( )
6
1
x
11. 2
x -1
3. 25.3 x > 81.5 x
12. 2
x 2 -2 x
4. 2 x -3 = 5 x
13. 3 x.8 x +1 = 36
2. 2 x
5.
3
2
-4
= 3 x -2
x -3
6. 5 x .8
7. 4.9
2
=5
x
x +1
x -1
x 2 - 7 x +12
14. 3 x.2
= 100
15. 3
2 x +1
2
9. x
1 1
+ lg x
10 5
10. x = 10
x
x +1
x -1
21. 9
æ1ö
=ç ÷
è2ø
-x
= 23
2x
x
[
x
17. 15
= x
18. 8
x- x2
x
x+2
2
22. 5 x .8
= 72
16. 57 = 7 5
9.2 2 x = 8 3 2 x +1
8.
.3 = 1,5
x
x
x
x +1 + x -1
lg x + 5
3
x
1
lg x
= 10 x
4
= 10 5+ lg x
x
-5 x + 6
= 3.2
20. 42.4 - 63.9 = 0
x
+ x-2
x -1
x
= 500 (KT’98)
23. x 3-log3 x = 900
x log 2 x < 32
x lg
x
x log 2 x
24. x lg x = 1000x 2
3
x
= 10
- log 22 x -3
= x2
x
]
( x-4)
25. x lg
=1
x + lg x - 4
2
> 10000
26. x (log3 x ) -3 log3 x = 3
3
= 4.3 4- x
27. 2
19. 5 x -1.3 x -2.7 x = 245
x +3
- 3x
2
+ 2 x -6
x log 2 x
8- 3 log 2
= 3x
2
2
2
-3
³
1
2
4
+ 2 x -5
- 2x
28. 2 x + 2 x - 2 + 2 x -1 = 7 x + 7 x -1 + 7 x - 2
VÊn ®Ò 6. Mét sè d¹ng kh¸c
L o¹i 1: Gi¶i bÊt ph-¬ng tr×nh:
1.
4x + 2x - 4
£ 2 (§HVH’97)
x -1
3.
32- x + 3 - 2 x
³ 0 (LuËt’96)
4x - 2
2.
21- x - 2 x + 1
£0
2 x -1
4.
31- x - 3 x + 2
£ 0 (Q.Y’96)
2x -1
L o¹i 2: B ×nh ph-¬ng
(
)
1.
2 5 x + 24 - 5 x - 7 ³ 5 x + 7
2.
2 13 x + 12 - 13 x - 5 ³ 13 x + 5
(
3.
8 + 21+
3- x
-4
3- x
+ 21+
3- x
>5
)
L o¹i 3: af(x) + af(x) . ag(x) (af(x) / ag(x) ) + ag(x) + b = 0. PP: § Æt af(x) = u, ag(x) = v.
1) 2 x
2
-5 x + 6
2) 4 x
2
+x
+ 21- x = 2.2 6-5 x + 1
2
+ 21- x = 2 ( x +1) + 1
2
2
3) 4 x
2
-3 x + 2
+ 4x
4) 4 x
2
- 2 x +1
+ 1 = 2 ( x +1) + 2 x
2
+ 6 x +5
= 42x
2
2
+3 x + 7
2
- 6 x +1
L o¹i 4:
1.
log 21 x
2
2
+x
log 1 x
2
>
5
2
2. x log x + 16 x - log x < 17
2
VÊn ®Ò 7.
2
Mét sè bµi to¸n chøa tham sè
1. T×m m ®Ó bÊtph-¬ng tr×nh cã nghiÖm:
T rang 6
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
1) 3 x ³ 1 + m 2
2
2) 3
x -1
£ 1 - m2
3) 5
-x
³ 1 + m2
1
4)
4
= 2m - 1
x -2
2. T×m m ®Ó c¸c ph-¬ng tr×nh sau cã nghiÖm :
1) 9 x + 3 x + m = 0
5) 2 x + (m + 1).2 - x + m = 0
2) 9 x + m.3x – 1 = 0
6) 16 x – (m – 1).2 2x + m – 1 = 0
3) 9 x + m.3 x + 1 = 0
7) 25 x - 2.5 x - m - 2 = 0
4) 3 2 x + 2.3 x - (m + 3).2 x = 0
8) 25 x + m.5 x + 1 - 2m = 0
1 x 2 - 4 x +3
( )
= m 4 - m 2 +1
5
3. V íi nh÷ng gi¸ trÞ nµo cña m th× ptsau cã 4 nghiÖm ph©n biÖt:
4. Cho ph-¬ng tr×nh: 4 x – (2m + 1)2 x + m 2 + m = 0
1
a) Gi¶i ph-¬ng tr× nh víi m = 1; m = 1; m = - .
2
b) T×m m ®Ó ph-¬ng tr×nh cã nghiÖm?
5. Cho ph-¬ng tr×nh:
c) Gi¶i vµ biÖn luËn ph-¬ng tr×nh ®· cho.
m.4 x – (2m + 1).2 x + m + 4 = 0
a) Gi¶i ph-¬ng tr×nh khi m = 0, m = 1.
c) T×m m ®Ó ph-¬ng tr×nh cã nghiÖm x Î [ -1; 1] ?
b) T×m m ®Ó ph-¬ng tr×nh cã nghiÖm?
6. (§HNN’98) Cho ph-¬ng tr×nh: 4 x – 4m(2 x – 1) = 0
a) Gi¶i ph-¬ng tr×nh víi m = 1.
b) T×m m ®Ó ph-¬ng tr×nh cã nghiÖm?
7. X ¸c ®Þnh a ®Ó ph-¬ng tr×nh:
8. T×m m ®Ó ph-¬ng tr×nh:
9. (§H CÇn Th¬’98)
(
c) Gi¶i vµ biÖn luËn ph-¬ng tr×nh ®· cho.
)
a. 2 x - 2 + 1 = 1 - 2 x cã nghiÖm vµ t×m nghiÖm ®ã.
m.4 x – (2m + 1).2 x + m + 4 = 0 cã 2 nghiÖm tr¸i dÊu .
Cho ph-¬ng tr×nh: 4 x – m.2 x + 1 + 2m = 0
a. Gi¶i ph-¬ng tr×nh khi m = 2.
b. T×m m ®Ó ph-¬ng tr×nh cã hai nghiÖm ph©n bi Ötx 1 , x2: x1 + x2 = 3.
10. V íi nh÷ng gi¸ trÞ nµo cña a th× ph-¬ng tr×nh sau cã nghiÖm:
11. T×m c¸c gi¸ trÞ cña k ®Ó ph-¬ng tr×nh: 9
12. T×m c¸c gi¸ trÞ cña a ®Ó pt:
x
7
- x +3
- 4.7
-
-m =0
– (k – 1).3 x + 2k = 0 cã nghiÖm duy nhÊt.
144 - úx - 1ú - 2.12 - úx - 1ú + 12a = 0 cã nghiÖm duy nhÊt.
1-
13. T×m c¸c gi¸ trÞ cña a sao cho ptsau cã
2 nghiÖm d-¬ng ph©n biÖt:
9
14. T×m c¸c gi¸ trÞ cña m ®Ó pt sau cã 2 nghiÖm x1, x2 tm: -1 < x1 < 0 < x2 :
15. (HVCNBCVT’99)
1
x +3
2
1
x2
1-
- a.3
1
x2
+2=0
m 2m + 1
+m+4=0
4x
2x
T×m c¶ c¸c gi¸ t rÞ cña m ®Ó bpt sau nghiÖm ®óng "x > 0
(3m + 1).12 x + (2 - m).6 x + 3 x < 0
T rang 7
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
ôcosx ô
16. T×m gi¸ trÞ cña tham sè a ®Ó bpt: 4
+ 2(2a + 1) 2
17. (GT’98) m.4 x + (m – 1).2 x + 2 + m – 1 > 0; "x
23. 7
18. (Má’98) 9 x – 2(m + 1)3 x – 2m – 3 > 0 ; "x
19.
(G T_TPHCM’99)
x
x
20. (D-îc HCM’99) 4 – m.2
x
21. 4 – (2m + 1).2
x+ 1
x+ 1
+ 3 + m < 0; "x
2
+ m + m ³ 0; "x
)
+m-2 4
1-tg 2 x
- 4.7
-
1
x +3
2
- m > 0 ; "x.
26. (GT_TPHCM’99) 9 x + m.3 x + 2m + 1 > 0 ; "x
2
27. 3 2x + 1 - (m + 3).3 x – 2(m + 3) < 0 ; "x
4 |cosx | + 2(2a + 1).2 |cosx | + 4a2 – 3 < 0
28. T×m mäi gi¸ trÞ cña m ®Ó bpt sau tho¶ m·n víi mäi x:
(m
- x +3
25. 4 sinx + 2 1 + sinx > m ; "x.
22. 25 x – (2m + 5).5 x + m 2 + 5m > 0 ; "x
29. T×m m ®Ó bpt:
+ 4a2 - 5 £ 0 nghiÖm ®óng víi mäi x.
24. 4 x – m.2 x + 1 + 3 – 2m < 0; "x
9 – m3 + 2m + 1 > 0 ; "x
x
ôcosx ô
- (m + 5)2
1-tg 2 x
- 2 £ 0 nghiÖm ®óng víi mäi x.
30. T×m c¸c gi¸ trÞ cña m ®Ó c¸c bÊt ph-¬ng tr×nh sau ®©y cã nghiÖm:
a. 3 2x + 1 – (m + 3).3 x – 2(m + 3) < 0
d. 3.4 x – (m – 1).2 x – 2(m – 1) < 0
b. 4 x – (2m + 1).2 x + 1 + m 2 + m ³ 0
e. 4 x + m.2 x + m – 1 £ 0.
c. 9 x – (2m - 1).3 x + m 2 - m ³ 0
f. m.25 x – 5 x – m – 1 > 0
31. T×m gi¸ trÞ cña m ®Ó cho hµm sè: f ( x ) =
32. Cho ph-¬ng tr×nh :
- x 2 + 3x - 3
(m - 1)æç 1 ö÷
è2ø
(5 + 2 6 ) + (5 - 2 6 )
tgx
a) Gi¶i ph-¬ng tr×nh víi a = 10 .
tgx
=a
+ 21+sin x + 2m
2
(§ 50)
b) Gi¶i vµ biÖn luËn pt theo a .
x
33. Cho ph-¬ng tr×nh:
nhËn gi¸ trÞ ©m víi mäi x
- cos 2 x
x
æ7+3 5ö
æ7-3 5ö
ç
÷ + aç
÷ = 8 (1)
ç
÷
ç
÷
2
2
è
ø
è
ø
a. Gi¶i ph-¬ng tr×nh khi a=7
34. (KTHN’99) Cho bÊtph-¬ng tr×nh :
b. BiÖn luËn theo a sè nghiÖm cña ph-¬ng tr×nh.
m .9 2 x
2
-x
- (2 m + 1 ).6 2 x
2
a)
Gi¶i bÊt ph-¬ng tr×nh víi m = 6.
b)
T×m m ®Ó bÊt ph-¬ng tr×nh nghiÖm ®óng víi mä i x mµ
-X
2
+ m .4 2 x
x ³
-X
£0
1
.
2
4 . Gi¶i ph-¬ng tr×nh: (D ïng tÝnh chÊt cña hµm sè - § o¸n nghiÖm?)
2 3 x + 2 2 x + (1 + 3 x 2 )2 x + x - 2 = 0
7. T×m m ®Ó ph-¬ng tr×nh cã nghiÖm : 3 mx ( 2- x ) = 4
VÊn ®Ò 8:
x -1
+m
HÖ ph-¬ng tr×nh mò
T rang 8
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
ì2 + 2 = 3
1. í
îx + y = 1
x
1
ì -2 x
2y
ï4 + 4 =
2. í
2
ïî x + y = 1
ì2 y = 200.5 y
3. í
îx + y = 1
ì x y 1
ï3 .2 =
4. í
9
ïî y - x = 2
ìï4 x + y = 128
5. í 3 x - 2 y -3
ïî5
=1
ìï27 x = 9 y
6. í x
ïî81 = 243.3 y
ìï64 x + 64 2 y = 12
7. í x + y
ïî64 = 4 2
ìï3 x + 3 y = 28
8. í x + y
ïî3 = 27
ìï3 x .5 y = 75
9. í y x
ïî3 .5 = 45
ì3 x - 2 y = 77
ï
í x
y2
2
ïî3 - 2 2 = 7
2
ì3 - 2 = 77
ï
10. í
y
ïî3 x - 2 2 = 7
2x
3y
ì
5
ï2 = 2 .2 x
í x
2( x - y )
ï y
y
3
=
3
.
3
î
y
y
ìï x + 2 = 3
í
ïî4 x + 4 y = 32
y +1
11
ì x
y
ïï3.2 + 2.3 = 4
11. í
ï2 x - 3 y = - 3
îï
4
2x
y
ìï7 - 16 = 0
12. í x
ïî4 - 49 y = 0
x
y
ìï x + 3 y -1 = 2
13. í
ïî3 x + 9 y = 18
ìï y 2 = 4 x + 8
í x +1
ïî2 + y + 1 = 0
ìï y 2 = 4 x + 2
14. í x + 2
ïî2 + 2 y + 1 = 0
ìï3 x 2 -5 x + 6 - log3 2 = 2 - y -1
27. í
2
ïî2 y - 2 - 5 y - ( y - 3) ³ -5
ìï5 x 2 -5 x + 4 -log5 2 = 2 y -3
29. í
2
ïî3 y - 5 y + 1 + ( y - 2 ) £ 3
y
ì x
16. ïí2 - 2 = y - x
ïî2 x 2 + 4 x - y 2 = -3
ìï3 2 x + 2 + 2 2 y + 2 = 17
17. í x +1
ïî2.3 + 3.2 y = 8
ìï9 2tgx +cos y = 3
; í cos y
1
- 81tgx = 2
î9
ïî(324 ) y = 2 x 2 ï
ìx =9
18. ïí
y
ì2 2 + 2 y £ 1
20. í
î x + y ³ -2
HN)
ìï4 x 2 -8 x +12 - log 4 7 = 7 2 y -1
28. í
2
ïî y - 3 - 3 y - 2( y + 1) ³ 1
ìï x + y = 2
15. í
2
ïî( x + 1) y + y + 2 = 1
xö
æ
ì y+4 x
5ç y - ÷
3ø
è
ïx
=
y
19. í
ïî x 3 = y -1
ì3 x 2 - 2 x -2 -log3 5 = 5 -( y + 4 )
ï
26. í
( SP
2
ïî4 y - y - 1 + ( y + 3) £ 8
ìï2 3 x +1 + 2 y -2 = 3.2 y +3 x
30. í
ïî 3 x 2 + 1 + xy = x + 1
(§HSPHN’98)
ìï3 x - 3 y = ( y - x)( xy + 2)
31. í 2
ïî x + y 2 = 2
ì2 x - 2 y = ( y - x )( xy + 2)
32. ïí
ïî x 2 + y 2 = 2
(KT’99)
ìï3 x .2 y = 972
í
ïîlog 3 9 x - y ) = 2
[(
( QG’95)
)]
ìcos p x 2 + y 2 = 1
ï 2
2
2
2
33. ïí4 x + (1+ y ) - 32 = 31 .2 x +(1+ y )
ïy ³ 0
ïî
ìï4 log3 ( xy ) = 2 + ( xy )log3 2
21. í 2
ïî x + y 2 - 3 x - 3 y = 12
ì x log8 y + y log8 x = 4
34. í
(TC’ 00)
îlog 4 x - log 4 y = 1
ì4 x + y -1 + 3.4 2 y -1 £ 2
22. í
î x + 3 y ³ 2 - log 4 3
35.
ìx 4 + 2 x x 2 - 6 x = 9
23. í
îx > 0
ì 12
-1
ïy = x
24. í
28 x - 7 y
ï( xy )x .x - y = y 2
î
36.
3x
ì
ïï x log 2 3 + log 2 y = y + log 2 2
í
ï x log 12 + log x = y + log 2 y
3
3
3
ïî
x
ìï2 3 x + y + 2 y -2 = 3.2 y +3 x
í
ïî 3 x 2 + 1 + xy = x + 1
1
ì 1
= (x + y )
ï
x- y
25. í 2 3
ï( x + y ).2 y - x = 48
î
T rang 9
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
ì1
ï .9 = 9
ìï x + y + a = 1
ï
1. Cho hÖ ph-¬ng tr×nh: í 3
Gi¶i theo a hpt:
í a 2 x + y - xy
ïî2 4.
=2
ï x + my = 2 x - 4
ïî x
y
a. Gi¶i hÖ ph-¬ng tr×nh víi m = 3,
b. T×m c¸c gi¸ trÞ cña m sao cho hÖ cã nghiÖm duy nhÊt. H ·y x¸c ®Þnh nghiÖm duy nhÊt ®ã.
1
y
x
2y
2. T×m a ®Ó hÖ sau cã nghiÖm víi mäi b:
ìï( x 2 + 1) a + (b 2 + 1) y = 2
í
ïîa + bxy + x 2 y = 1
3. X ¸c ®Þnh a ®Ó hÖ cã nghiÖm duy nhÊt:
ìï2 x + x = y + x 2 + a
í 2
ïî x + y 2 = 1
4. Cho hÖ ph-¬ng tr×nh:
ìïlog 3 x 2 + 2 log y = 0
í 3
2
ïî x + y + my = 0
a. Gi¶i hÖ pt khi m = 1.
b. V íi m=? th× hÖ cã nghiÖm > 0
ìï9 x - 4 y 2 = 5
(1)
í
ïîlog m (3 x + 2 y ) - log 3 (3 x - 2 y ) = 1
2
5. Cho hÖ ph-¬ng tr×nh:
a. Gi¶i hÖ ph-¬ng tr×nh (1) víi m = 5.
6. Cho hÖ ph-¬ng tr×nh:
b. T×m m ®Ó hÖ (1) cã nghiÖm (x,y).
1
ì x
y
ïa + a =
2
í
ïx + y = b 2 - b + 1
î
b. T×m a ®Ó hÖ cã nghiÖm víi mäi x Î [0;1]
a. Gi¶i hÖ ph-¬ng tr×nh víi b =1 vµ a > 0 bÊt k×.
7. Cho bÊt ph-¬ng tr×nh:
a. Gi¶i bpt víi m=4
x+4 1
3
T rang 10
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
Ph-¬ng tr×nh – bÊt ph-¬ng tr×nh l«garit
v Ph-¬ng h¸p ®-a vÒ cïng mét c¬ sè
V Ý dô. Gi¶i ph-¬ng tr×nh:
log3x + log9x + log27x = 11 (1)
Gi¶i:
(! ) Û log x + log
§ -a vÒ c¬ sè 3, ta ®-îc:
3
Ûlog3x = 6 Û x = 3 6 = 729.
32
x + log
1
1
x = 11 Û log 3 x + log 3 x + log 3 x = 11
33
2
3
V Ëy ph-¬ng tr×nh ®· cho cã nghiÖm lµ x = 729.
B µi tËp t-¬ng tù
1.
log 3 (log 2 x) = 1
2.
log 2 ( x 2 + x - 3) 2 = 0
æ1
ö
17. log 1 ç log 3 x - 2 ÷ = 0
3
ø
5è
3.
log2(x2 – 4x – 5) £ 4
18. log x 10 2 = -0,01
4.
log12(6x2 – 4x – 54) £ 2
19. log 2 x + log 4 x = log 1 3
5.
log 1 log 4 x - 5 > 0
(
)
2
6.
log3(5x + 6x + 1) £ 0.
7.
log 1 1 + x - x - 4 £ 4
8.
2
(
log 1
)
2
(x
2
)
- 2 - x +1 £ 0
1ö
æ
log 3 ç x 2 - 9 - x + ÷ ³ -1
3ø
è
10. log2(25 x + 3 –1) = 2 + log2(5 x + 3 + 1)
11. logx(2x2 – 7x +12) = 2
12. log3(4.3 x – 1 ) = 2x – 1
log 2 x ( x + 62 ) = 3
21. log 3 (x + 1) + 2 = log 3x
2
13. log2(9 - 2 ) = 3 – x
14. log2x – 3 16 = 2
log 3
log2x – 3x = 2
2x - 3
< 1 (SPVinh’98)
1- x
16. 2 log 5 5 - 2 = log x
1
5
3log3 x – log9 x = 5
logx + 1(3x2 – 3x – 1) = 1
22. log(2 (x – 1) + log2 x = 1
23. log 3
x +1
x
;
= log 3
x
2- x
24. lg x =
25.
(
1
lg( x + 1) ;
2
log 5 ( x - 1) = log 5
log 4
)=3 ;
lg x + 1 + 1
lg 3 x - 40
(
(
26. lg 10 lg x
x
15.
log2(|x+1| - 2) = - 2
2
5
9.
log 7 (2 x 2 - 5 x + 13) = 2
20. log2(4.3 x - 6) - log2(9 x - 6) = 1
2
2
æ
ö
1
÷ =1
log 2 çç
÷
x
1
1
è
ø
27. 2
(
2
- 21
x
x +1
2x - 1
1
< - (§ HVH’98)
x+2
2
log 5 ( x - 1) = log 5
x
x +1
)
) - 2 = lg x - 2 lg 5
log 8 x 2 - 6 x + 9
) = 32 log
x
x -1
log (1x
3
2
+3 x -4)
= log (12 x + 2 )
3
28. log4(log2x) + log2(log4x) = 2
29. logx + 1 (2x3 + 2x2 – 3x + 1) = 3
30. log2x.log3x = log2 x2 + log3x3 – 6
T rang 11
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
31. log 7
2
49. 2 + lg(1 + 4x – 4x) – lg(19 + x2) = 2lg(1 – 2x)
2x + 3
2
+ log 1
=0
21
3
x
6
7
32. log 3 (1 - x ) + log 1
3
æ
è
50. 2 lgç x +
6
=0
2-x
æ
è
51. lgç x +
2
3
33. log3x.log9x.log27x.log81x =
1
3
4 log 2 1 - 2 x 2
(
2logx - 116
53. log(2x
+3 x + 2 )
)
4
2
+3 x +12)
= 3 + log32
x 2 + 4x + 4 = 9
3
55. log 5 x + 3 log 25 x + log
3
8
2
125
11
2
x
x
x
57. log 3 + log 3 + log 1 = 6 log x 2. log 2 x 2 = log 4 x 2
3
3
2
58. log2x + log4x + log8x = 11.
39. log 2 x + log 4 x + log 8 x =
x3 =
2a - x
- log 1 x = 0
a
a
a
(QGHN’99)
38. log 2 x + log 4 x = log 1
+ log(2x
54. log 3 ( x + 2 ) + log 3
56. log
x + 8x - 1
£2
x +1
2
2
36. log 2 x = 1 + 3 log 2 3 - 3 log 2
37. log 2
4ö
1ö 1
1
æ
÷ - lgç x - ÷ = lg(x + 6 ) - lg x
3ø
3ø 2
2
è
52. (x – 4) 2 log4(x – 1) – 2log4 (x – 1) 2 = (x – 4) 2logx-1 4 –
x
æ xö
34. log 1 ç1 - ÷ + log 2 2 - = 0
2ø
4
2è
35. log1- 2 x 2 x =
1ö
5ö
æ
÷ - lg (x - 1) = lgç x + ÷ + lg 2
2ø
2ø
è
11
2
40. log 3 x + log 9 3 x + log 27 x =
59. log (xx
2
+ 4 x -3 )
=3
;
log2x – log16 x = 3
log 1 (log x1 ) = -1
3
2
5 æ 11 ö 60. lg5 + lg(x + 10) = 1 – lg(2x – 1) + lg(21x – 20)
ç ÷
3 è 12 ø
3
41. log2(x + 3) + log2(x – 1) = log2 5
x
1
æ3ö
61. log 3 ç ÷. log 2 x - log 3
= + log 2 x
3 2
èxø
42. 1 + 2 log x 2. log 4 (10 - x ) =
62. log 3 x + log 3 ( x + 2) = 1 ; x(lg5 – 1) = lg(2 x + 1) – lg6
æ
è
43. ç1 +
2
log 4 x
æ
1 ö
÷ lg 3 + lg 2 = lgçç 27 - 3
2x ø
è
1
x
ö
÷
÷
ø
2ö
1
æ
44. log 2 x 2 -1 ç x 2 - ÷ = 2 3ø
log 3 2 x 2 - 1
è
(
)
45. lg(2 x - 3) 2 - lg(3x - 2) 2 = 2
46. lg x + lg( x - 1) = lg(5 - 6 x) - lg 2
47. log 4 x + log16 x + log 2 x = 7
63. log 4 log 2x + log 2 log 4x = 2 ; x + lg(1 + 2 x ) = x lg 5 + lg 6
1
64. 3 +
log 32
æ 75 x 11 ö
= log x ç
- ÷
x
4
2ø
è
2
2
2
65. log 2 x 2 + 6 x +8 log 2 x 2 + 2 x + 3 ( x - 2 x ) = 0 ;
x
66. log 2x + log 3x + log (4x +1) = log10
1
æ 32
ö
- 16 x ÷ =
-3
è x
ø log 56 2 x
67. log 2 x ç
48. 1+lg(1+x2 – 2x) – lg(1 + x2) = 2lg(1 – x)
T rang 12
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
68.
75. log 4 {2 log 3 [1 + log 2 (1 + 3 log 2 x )]} =
1
log 2 (5 - x ) + 2 log 8 3 - x = 1
3
1
2
69. log4log3log2x = 0 logplog2log7 x £ 0
76. lg5 + lg(x + 10) = 1 – lg(2x – 1) + lg(21x – 20)
1 + 2 log 9 2
70.
- 1 = 2 log x 3. log 9 (12 - x)
log 9 x
77. log2(x2 + 3x + 2) + log2 (x2 + 7x + 12) = 3+ log23
78. 2log3(x – 2) 2 + (x – 5) 2logx – 2 3 = 2logx – 2 9 +
(x – 5) 2log3(x – 2)
1
71. log 4 {2 log 3 [1 + 3 log 2 x]} =
2
3
(3
72. ( x - 1) log 5 + log 5
x +1
+ 3)
79. log 9 (x 2 - 5 x + 6 ) = 2 -1. log
2
= log 5(11.3
x
-9 )
73. log 3 ( x + 1) + log 4 ( x + 1) = log 5 ( x + 1)
74. log a {1 + log b [1 + log c (1 + log d x ]} =
80. log 2 log 1 log 5 x > 0 ;
3
x -1
+ log 3 x - 3
2
log 4 log 2 log x 5 =
3
1
2
1
2
v Ph-¬ng ph¸p ®Æt Èn sè phô
L o¹i 1:
V Ý dô.
Gi¶i ph-¬ng tr×nh:
1
2
+
=1
5 - lg x 1 + lg x
Gi¶i:
§ Ó ph-¬ng tr×nh cã nghÜa, ta ph¶i cã: lgx ¹ 5 vµ lgx ¹ -1.§ Ætlgx = t(* ) (t ¹ 5 , t ¹ 1), ta ®-îc pt Èn t:
1
2
+
= 1 Û 1 + t + 2(5 - t ) = (5 - t )(1 + t )
5 - t 1+ t
é
êt =
2
2
Û 1 + t + 10 - 2t = 5 + 5t - t - t Û t - 5t 2+ 6 = 0 Û ê
(D =( -5) -4.6=1)
êt =
ëê
5 -1
=2
2
5 +1
=3
2
Ta thÊy 2 nghiÖm trªn ®Òu tho¶ m·n ®iÒu kiÖn cña t. D o ®ã:
+ V íi t = 2, thay vµo (* ) ta cã: lgx = 2 Û x = 10 2 = 100.
+ V íi t = 3, thay vµo (* ) ta cã: lgx = 3 Û x = 10 3 =1000.
V Ëy ph-¬ng tr×nh ®· cho cã 2 nghiÖm x = 100 vµ x = 1000.
B µi tËp t-¬ng tù.
L o¹i 1:
1.
4. log 24 x + 2 log 4 x 2 + 1 = 0
3.
log 3x 10 + log 2x 10 - 6 log10x = 0
2.
log x 5 5 - 1,25 = log 2x 5
4.
log 2 (5 x - 1). log 24 (5 x - 1) = 1
T rang 13
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
5.
log 2 (2 x) . log 2 = 1
27. log 2 ( x - x 2 - 1). log 3 ( x + x 2 - 1) = log 6 x - x 2 - 1
6.
log 2 (3 x + 3) - 4. log 3x +3 2 = 0
28. log
log 2 2 + log 2 4 x = 3
29. 2 2 log3 (x
7.
2
2
x
x
8.
3 lg x 2 - lg 2 (- x) = 9
9.
lg x 2 + 9 lg 2 x = 40
10. log x 3. log x 3 + log x 3 = 0
3
81
11. log 3 (4 x + 1) + log 4 x +1 3 >
5
2
1
12. log 7 x - log x = 2
7
14. logx2 – log4x +
7
=0
26
2
x
22. 4
- 6.2
log 22 x +1
x
- 11 = 0
4
2. log 22 x - 3. log 2
31. 2 log 5x - log125
<1
x
lg 22 x 3 - 10 lg x + 1 > 0
;
32. log 22 ( x + 1) 3 - log (2x +1) - 4 < 0 ;
3 lg x 2 - lg 2 (- x) 2 = 9
33. lg(10 x). lg(0,1x) = lg x 3 - 3 ;
4 log 24 (- x) + 2 log 4x + 1 < 0
34. log 3 a - log x a = log x a ;
8lg3 x – 9lg2x + lgx = 0
2
+2
log3 27
3
36. log a (ax ). log x (ax ) = log a 2
(
37. 1 + log 27 x
log 27 x
) = 103 log
1
a
27
38. lg4(x – 1) 2 + lg2(x – 1) 3 = 25
41.
x
(Y HN’00)
x
(
)
(
log 9 3x 2 - 4 x + 2 + 1 > log 3 3x 2 - 4 x + 2
(
=0
23. 4 log3 x - 5.2 log 3 x + 2 log 3 9 = 0
24. 2
) = 24 3 log x - log 3 x - 1 = 0
3
3
-16 +1
)
(
)
(SPHN’00)
)
42. log 22 x - x 2 + 2 + 3 log 1 x - x 2 + 2 + 2 £ 0
2 lg x
2
= - lg x +
lg x - 1
lg x - 1
log9 x
2
30. 5 2 (log 5 2+ x ) - 2 = 5 log 5 2 + x ;
9
20. log 2 (2 x ) . log 2 < 1
log9 x
3
3
2
40. 5. log x x + log 9 x + 8 log 9 x 2 x = 2
x2
2
19. 2log3(2x + 1) = 2.log2x + 1 3 + 1
21.
) + 2 log (x
39. 2log4(3x – 2) + 2.log3x – 2 4 = 5
17. log 2 x 64 + log x 2 16 = 3
2
-16
3
1
2
16. (log 5 x ) + log 5 5 x - 2 = 0
2
18. log x + 3 ³ 2 log
2
35. log 1 x - 2 + 3 = log 1 x + 1 ;
15. 5. log 2 x - log 2 4 x - 4 = 0
2
2
log2|x + 1| - logx + 1 64 = 1
2 + 4 log 4 x 2 + 9 = 0
3
5
2
13. log2x + logx2 =
x
= x 2 log 2 x - 48
2
43. log 2
x - 4 log 4 x - 5 = 0 (C§SPHN’97)
44. log3x + 7(9 + 12x + 4x2 ) + log2x + 3(21 + 23x + 6x2 ) = 4
45. log1-2x(1 - 5x + 6x2 ) + log1 - 3x(1 - 4x + 4x2) = 2
46. log 2 ( x - x 2 - 1). log 3 ( x + x 2 - 1) = log 6 x - x 2 - 1
log x +1
2 log x
25. 2 2 + 224 = x 2
2
26. log 22 (2 - x ) - 8 log 1 (2 - x ) ³ 5
4
L o¹i 2: § «i khi ®Æt Èn phô nh-ng ph-¬ng tr×nh vÉn chøa Èn ban ®Çu.
T rang 14
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
1.
lg x - lg x. log 2 4 x + 2 log 2 x = 0
5.
( x + 2) log ( x + 1) + 4( x + 1) log 3 ( x + 1) - 16 = 0
2.
log 52 ( x + 1) + ( x - 5) log (5x +1) - 16 = 0
6.
( x + 3) log 32 ( x + 2) + 4( x + 2) log 3 ( x + 2) - 16 = 0
3.
log 22 x + ( x - 4) log 2x + 3 - x = 0
7.
log22x + (x – 1)log2x + 2x – 6 = 0. (TS’97)
4.
log 22 x + ( x - 5). log 2 x - 2 x + 6 = 0
2
2
3
L o¹i 3:
1. log 22 x - log 2x + log 3x - log 2x . log 3x = 0
2. 2 log 2 ( x 2 - x) + log 2x - log 2x . log (2x
2
- x)
=2
L o¹i 4:
4.6) log 32 x + 2 = 33 3 log 3x - 2
4.1) log 2 ( x - x 2 - 1) + 3 log( x + x 2 - 1) = 2
4.2)
3
2 - lg x = 1 - lg x - 1
4.7) log 22 x + log 2 x + 1 = 1
4.3)
3
1 - log 3x + 3 1 + log 3x = 1
4.8) 6 x = 3 log (65 x +1) + 2 x + 1
4.4)
3 + log (4x
2
-4 x)
+ 2 5 - log (4x
2
-4 x )
4.9) 7 x -1 = 6. log (76 x -5) + 1
=6
4.5) x + 1 - lg 2 x = 10
v Ph-¬ng ph¸p sö dông tÝnh ®¬n ®iÖu cña hµm sè l«garit
L o¹i 1:
2. log 1 x = x - 4
log2(3x – 1) = -x + 1
4. 2 x + log x1 = 5
3. x + log 3x = 4
2
3
L o¹i 2: Ph-¬ng tr×nh kh«ng cïng c¬ sè.
V D 1. Gi¶i ph-¬ng tr×nh:
1. 2
log 5( x + 3 )
=x
2. 2 log 5 ( x +3) = x
( x -3)
3. 3log 2
=x
4. log 3x £ log 7x
5.
log (2x +1) ³ log 52 x
( )
log (1 + x ) = log
9. 2 log 6 ( 4 x + 8 x ) = log 4 x
6. log 2 1 + x = log 3 x
7.
2
3
8. log 3 x + 1 > log
2
10. log 2
x
(
)
x+ x =
4
1
log 3 x
4
11. 2. log 3 cot x = log 2 cos x
( -2 x - 2 )
2
L o¹i 3: f(x) = f(y) Û x = y, (f - ®ång biÕn hoÆc nghÞch biÕn )
1. log3(x2 + x + 1) – log3 x = 2x – x2.
2. 2 x - 21- x = log 2
1. T×m k ®Ó ph-¬ng tr×nh cã ®óng 3 nghiÖm: 4
- x -k
. log
2
(x
2
1- x
x
)
- 2x + 3 + 2-x
2
+2 x
. log 1 (2 x - k + 2 ) = 0
2
T×m m ®Ó ph-¬ng tr×nh cã nghiÖm. 1) log 3 2 ( 4 - x + x + 5 ) = a ;
2) log 2 (4 x + 4 2 - x ) = a
L Ëp b¶ng xÐtdÊu:
T rang 15
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
1.
log 2 ( x + 1) - log 3 ( x + 1)
>0
x 2 - 3x + 4
3.
log 2 ( x + 1) - log 3 ( x + 1)
>0
x 2 - 3x - 4
2.
log 2 ( x + 1) - log 3 ( x + 1)
>0
x 2 + 3x - 4
4.
log 2 ( x + 1) - log 3 ( x + 1)
>0
x 2 + 3x - 4
2
3
2
2
3
3
Ph-¬ng tr×nh l«garit chøa tham sè
1. T×m c¸c gi¸ trÞ cña m ®Ó ph-¬ng tr×nh sau cã hai nghiÖm ph©n biÖt:
a. log3 (9 x + 9a3) = 2
b. log2(4 x – a) = x
2. (§H’86) T×m m ®Ó ptsau cã 2 nghiÖm tm x 1, x2 tm : 4 < x1 < x2 < 6:
(m - 3) log 21 (x - 4) - (2m + 1) log 1 (x - 4) + m + 2 = 0
2
2
3. T×m c¸c gi¸ trÞ cña a ®Ó ph-¬ng tr×nh sau cã 2 nghiÖm tho¶ m·n: 0 < x
1
< x2 < 2:
2
(a – 4)log2 (2 – x) – (2a – 1)log2(2 – x) + a + 1 = 0
nghiÖm x1, x2 tho¶ m·n x 12 + x22 > 1:
4. T×m c¸c gi¸ trÞ cña m ®Ó ph-¬ng tr×nh sau cã hai
2log[ 2x2 – x + 2m(1 – 2m)] + log1/2(x2 + mx – 2m 2) = 0
5. (§HKT HN ’98) Cho ph-¬ng tr×nh:
(x - 2)log
2
4( x-2)
= 2 a ( x - 2) 3
a. Gi¶i ph-¬ng tr×nh víi a = 2
b. X ¸c ®Þnh c¸c gi¸ trÞ cña a ®Ó pt cã 2 nghiÖm ph©n biÖtx 1, x2 tho¶ m·n:
5
£ x1 , x 2 £ 4 .
2
6. V íi gi¸ trÞ nµo cña a th× ph-¬ng tr×nh sau cã nghiÖm duy nhÊt:
a. log 3 ( x + 3) = log 3 ax
e. lg(x2 + 2kx) – lg(8x – 6k – 3)
b. 2lg(x + 3) = 1 + lgax
f.
(
)
lg(2 x - a - 1) + log 1 x 2 + 4ax = 0
10
c. lg(x2 + ax) = lg(8x – 3a + 3)
g. log
lg kx
d.
=2
lg( x + 1)
h. log 2
5 +2
(x
2+ 7
2
)
+ mx + m + 1 + log
(x - m + 1) + log 2
5 -2
2- 7
x=0
(mx - x 2 ) = 0
7. T×m c¸c gi¸ trÞ cña m sao cho ph-¬ng tr×nh sau nghiÖm ®óng víi mäi x:
(
)
(
)
log âg 2 + 2 3 - m - 1 - log 2 m 3 x 2 - 5m 2 x 2 + 6 - m = 0
8. T×m c¸c gi¸ trÞ cña m ®Ó hµm sè sau x¸c ®Þnh víi mäi x:
y = 2 log3 [( m+1) x
2
]
- 2 ( m -1) x + 2 m -1
a ö 2
a ö
a ö
æ
æ
æ
9. ç 2 - log 2
÷ x + 2ç1 + log 2
÷ x - 2ç1 + log 2
÷ > 0 ; "x
a +1ø
a +1ø
a +1ø
è
è
è
10. (AN’97) log2 (7x2 + 7) ³ log2(mx2 + 4x + m) ; "x
T rang 16
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
11. (QG TPHCM’97) 1 + log5 (x + 1) ³ log5 (mx + 4x + m) ; "x
2
12. log
1
m -1
(x
2
2
)
+ 2 m > 0 ; "x
13. T×m c¸c gi¸ trÞ cña m sao cho kho¶ng (2; 3) thuéc tËp nghiÖm cña bÊtph-¬ng tr×nh sau:
log5(x2 + 1) ³ log5(m x2 + 4x + m) - 1
x 2 - 2 x log 1 a 2 + 3 - log 1 a 2 < 0
14. V íi gi¸ trÞ nµo cña a th× bptsau cã Ýt nhÊtmétnghiÖm:
2
15. V íi gi¸ trÞ nµo cña m th× bpt: log
m + 2
2
(2x + 3) + logm + 1(x + 5) > 0 ®-îc tho¶ m·n ®ång thêi t¹i
x = -1 vµ x = 2.
16. Gi¶i vµ biÖn luËn thep tham sè a c¸c bÊt ph-¬ng tr×nh sau :
a. loga(x – 1) + logax > 2
c.
b. loga(x – 2) + logax > 1
loga(26 – x2 ) ³ 2loga(4 – x) (HVKTMËt m·’98)
2
35. (NN’97) BiÕt r»ng x = 1 lµ mét nghiÖm cña bÊt ph-¬ng tr×nh:m (2x
log
+ x + 3) £ logm (3x2 – x).
H ·y gi¶i bÊtph-¬ng tr×nh nµy.
Mét sè ph-¬ng tr×nh, bÊt ph-¬ng tr×nh mu vµ l«garit liªn quan tíi l-îng gi¸c
1. (§ H K TH N ) T×m tÊt c¶ c¸c nghiÖmthuéc ®o¹n
1
2
1
+ log 5 sin x
2
1
2
1
+ log 5 cos x
2
2. 5 + 5
3. 6 + 3
4.
5.
1
3x - 5
1
2x - 1
= 15
=9
= (3 x - 5)
1
+ log15 cos x
2
10. log 2 tan x + log 4
6. log 7 - x 2
1
+ log 9 sin x
2
(
log 1 2 + 5 x - x 2
= (2 x - 1)
é 3 5ù
ê- 4 ; 2 ú cña ph-¬ng tr×nh:
ë
û
(
log 1 1+ 7 x - 2 x 2
)
4
2
x
=3
3 sin 2 x - 2six
= log 7 - x 2 2
sin 2 x. cos x
7.
( 3)
-
8.
2 cos 2 x -
1
=0
2.2 cos 2 x
9.
3.log22sinx + log2(1 – cos2x) = 2
tg 2 x
)
25
4 cos 2 x + 4 cos
3 3
=0
3tg 2 x
cos x
9
=0;
£ x£3
2 cos x + sin x
4
11. T×m c¸c cÆp (x, y) tho¶ m·n c¸c ®iÒu kiÖn:
ì
pö
æ
ïlog 2 (3 - sin xy ) = cosç px - ÷
6ø
è
í
ï2 £ x £ 3; 2 < y < 5
î
3p
2 æ ax
+
12. T×m aÎ(5; 16), biÕtr»ng PT sau cã nghiÖm thuéc [ 1; 2] : 1 + cos ç
8
è 2
ö æ1ö
÷=ç ÷
ø è3ø
cos px -sin x
T rang 17
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
é
5p
æp
log 3 ê1 + sin 2 ç x +
2
è2
ë
13. T×m aÎ(2; 7), biÕt r»ng PT sau cã nghiÖm thuéc [ 1; 2] :
öù
÷ú = cos ax - 1
øû
æ
æ
x
3 3ö
x
3 3ö
÷ + log 6 ç sin - 3 tan 2 x ÷=0
14. log 1 çç sin - 3 tan x ÷
ç
÷
2
2
2
2
6è
ø
è
ø
æ
æ
x
3 3ö
x
3 3ö
÷ + log 5 ç cos + 3 tan x ÷=0
15. log 1 çç cos + tan 2 x ÷
ç
÷
2
2
2
2
5è
ø
è
ø
x
x
æ
ö
æ
ö
16. log 1 ç sin + cos 2 x ÷ + log 3 ç sin - sin x ÷ = 0
2
2
ø
è
ø
3è
17. (HVKTQS’97) log 6 x - x 2 (sin 3 x + sin x ) = log 6 x - x 2 (sin 2 x )
10
10
BÊt ph-¬ng tr×nh mò vµ l«garit
B µi 1. Gi¶i c¸c ph-¬ng tr×nh vµ bÊt ph-¬ng tr×nh sau:
æ1ö
1. ç ÷
è2ø
4 x 2 -15 x +13
æ1ö
<ç ÷
è2ø
4 -3 x
2. 5 x – 3 x + 1 > 2(5 x – 1 – 3 x – 2)
3. 7 x – 5 x + 2 < 2.7 x – 1 – 118.5 x – 1 .
4. 5
x 2 - 7 x +12
> 1;
log
1
x -1
1ö
æ
5. log x ç x - ÷ ³ 2
4ø
è
6. log x
3
(5x
2
9.
)
(x
2
)
- 4 . log 1 x > 0
2
x + 1 + x - 1 - 2 . log 2 ( x - x) = 0
2
10. log5[ (2 x – 4)(x2 – 2x – 3) + 1] > 0
11. log x -
2
x+7
£ log x x-2
æ1ö
12. 2 x -1 > ç ÷
è 16 ø
1
x
; 3
2
-3 x - 4
=6
2
; 5
;
9
+ 2 x -6
æ1ö
<ç ÷
è 3ø
19. log x -
2
= 3x
2
x+7
£ log x x-2
20. 5 1 + x – 5 1 – x > 24
21. 2
2 x +1
æ1ö
- 21.ç ÷
è2ø
æ1ö
22. 3 4-3 x - 35.ç ÷
è 3ø
2x
1
4 x 2 -3 x +
2
2
(BKHN’97)
2
x+2
log 3
-x
<1
æ1ö
=ç ÷
è2ø
+ 2 x -5
x +1+ x -1
- 2x
18. 7 x – 5 x + 2 < 2.7 x – 1 – 118.5 x – 1 .
(4x -16x +7).log2(x – 3) > 0
(
x
x+2
< 3x
x - x -1
17. 5 x – 3 x + 1 > 2(5 x – 1 – 3 x – 2 )
)
8.
-3 x - 4
æ1ö
³ç ÷
è3ø
16. 2 x + 3 - 3 x
- 18 x + 16 > 2
2
2
15. 3 .8
(§ H H uÕ_98)
æ 2 5
ö
x + 1÷ ³ 0
3x ç x 2
ø
x 2 +1 è
14. 2 x
x
0,4 > 0
7. log
13. 3
x2 -2 x
-40 x 2
2
2x
9 x – 2.3 x – 15 > 0
2 x +3
+2³0
2 -3 x
+6³0
23. 8 lgx – 19.2 lgx – 6.4 lgx = 24 > 0
24. 5.36 x – 2.81 x – 3.16 x £ 0.
25. 25 - x
2
+ 2 x +1
+ 9-x
2
+ 2 x +1
³ 34.15 - x
2
+2 x
T rang 18
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
æ
ö
26. log 3 çç log 1 x - log 2 x + 2 ÷÷ < 1
4
è
ø
27. 6
log 62 x
45. 2(lg x) + (1 - 2 ) lg x 2 = 2 2
2
46. (log 2 x) 2 + 3 = 2(1 + 3 ) log 2 x
lg( x + 1 + 1)
+ x log 6 x £ 12 ;
lg x - 40
3
=3
11.3 x -1 - 31
28.
³ 5 ; logx(8 + 2x) < 2
4.9 x - 11.3 x -1 - 5
30.
31.
5
4
48. log 5 (6 - x) + 2 log
(6 - x) + log 3 > 270
1
5
lg( x 2 - 1)
<1
lg( x - 1)
49. log 4 3 x - log 2 x > 2 ;
4 - 7.5
2
£
x
- 12.5 + 4 3
x
29.
47. log 2 (2 - x) - 8 log 1 (2 - x) ³ 5
2 x +1
log
2
( x - 3)
x 2 - 4x - 5
x-5
³0
log 2 ( x - 4 ) - 1
³0 ;
(
lg(2 x + 4 )
=2 ;
lg(4 x + 7 )
)
lg x 2 - 3 x + 2
>2
lg x + lg 2
2
)
- 2 - x +1 £ 0 ;
5
(
log 1
2
)
3x + 1
³ -1
x +1
36. 3lg x + 2 < 3 lg x
2
+5
-2 ;
55. log 5 (1 - 2 x) < 1 + log 5 ( x + 1) ;
log 1 x = x - 6
2
3ö
3ö
æ
÷ + log 2 ç x - ÷ = 3
xø
xø
è
38. log 1 x + log 4 x ³ 1
(SPHN’94)
log x 2 (3 - 2 x ) > 1
5
39. log
2 x -1
2 x - 3 = 2 log 8 4 + log 2
2
æ
3x - 1 ö
x +1 ö
æ
÷
40. log 3 ç log 4
÷ £ log 1 çç log 1
÷
x +1 ø
3
x
1
è
3è
4
ø
3
41. log x
> -2
8 - 2x
(
log x 2
2x
1
£
x-3 2
)
log a 35 - x 2
42.
> 0 (0 < a ¹ 1)
log a (5 - x )
43. log3(3 x -1).log3(3 x + 1 – 3) = 6
44. x + lg(x2 – x – 6) 4 + lg(x + 2)
2
58. log 2 2 x + 3 log 2 x + log 1 x = 2
2
59. log 4 log 2 x + log 2 log 4 x = 2
60. log
(6 x +1 - 36 x ) ³ -2
1
5
1
3
3
x - 1)
2
57. log 1 (4 - x) 2 > log 1 (6 x - 3)
5
æ
37. log 2 ç x +
è
x>6
x>4
7
56. log 3 ( x 2 - 2) < log 3
35. log 2 2 - x - x - 2 ³ 1
2
x - 2 log
52. 2 log 7
5
54. 3 log 2 3 x - 4 log 4 x > 2 ;
33. log2(x + 1) – logx + 1 64 < 1
(x
51. 15 log 5 5 x - 2 log
53. lg 1 + x + 3 lg 1 - x = lg 1 + x 2 + 2
32. log8(x – 2) – 6log8(x – 1) > -2.
34. log 1
50. log 3 x - 2 log 9 x > 2
61. log 2 (4 x + 4) = x + log 2 (2 x +1 - 3)
62. log 1 (4 - x ) ³ log 1 2 x - log 1 ( x - 1)
2
2
2
63. 2 - log 2 ( x 2 + 3 x) ³ 0
64. 3 log 2 2 x - 2 - 9 log 2 x + 2 = 0
65. log 2 log 3 x - 3 ³ 0
(§ H T huû L îi 97)
3
66. x
log
x
2x
=4 ; 4
log 1 ( 2 x + 3)
2
2
1
= ( ) log3 ( 2 x -1)
3
T rang 19
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò
1
67. log x ( x - ) ³ 2
4
;
log x 20 - x > 1
1
75. ( )
9
68. log 2 (lg x + 2 lg x + 1) - 2 log 4 ( lg x + 1) = 1
71. log 3 x - x 2 (3 - x) > 1
1
ù
x +1 - log 3 ( x 2 -1) ú
2
û
76. log x ( x 2 -
69. log 5 ( x + 5) + 2 log 5 1 - 3 x > 1
70. log 2 x 64 + log x 2 16 ³ 3
é
ê log 3
ë
= 2( x - 1)
1
x) > 1 ;
2
log 1 (1 - x ) >
2
77. 1 + 2 log x 2. log 4 (10 - x) =
(§ H Y H N )
(§ H D L 97)
1
2
2
log 4 x
78. log 22 (2 + x - x 2 ) + 3 log 1 (2 + x - x 2 ) + 2 £ 0
72. log x (5 x 2 - 8 x + 3) > 2 (§ H V ¨n L ang)
2
79. log 9 x 2 (6 + 2 x - x 2 ) ³ 4 ; log x 20 - x > 1
73. log x 3 (5 x 2 - 18 x + 16) > 2
80. log x +1 ( x 2 + x - 6) 2 ³ 0 ;
2
2x +
5 > 0 ; log 4 x + 1 < 0
74. log x
x
6( x + 1)
5(1 - x)
9
x 2 -3
+3<3
x 2 -3
.28
B µi 2. Gi¶i c¸c ph-¬ng tr×nh vµ bÊt ph-¬ng tr×nh sau:
1. x 2 log x 27. log 9 x > x + 4
x
-x
2. (2 + 3.2 )
3.
2 log 2 x -log 2 ( x + 6 )
12.
>1
x-5
³0
log 2 ( x - 4) - 1
é
ù
x2
4. log 3 êlog 1 ( + 2 log 2 x -1 ) + 3ú £ 0
2
2ë
3
û
6. log x 125 x. log x < 1
7. log 4 . log 2 x + log 2 . log 4 x > 1
8. 5
log 1 log 2 .32 log 3 3
2
16
14.
x
2
10. log x 2. log 2 x 2 > log 4 x 2
3
lg( 2 x + 4)
=2 ;
lg 4 x + 7
2
log 22 x + ( x - 1) log 2 x = 6 - 2 x
15. 3 log3 x - 9 - 2 x log3 x = 0 ;
2
log 9- x 2 cos x. log 1 (9 - x 2 ) > 1
2
16.
1
1
<
log 3 ( x - 7 x + 12) log 3 20
2
17. log
- 3x + log 3 9 < 1
9. log 2 log 1 (2 - 4 ) £ 1
x
log 2 x
log 8 4 x
=
log 4 2 x log16 8 x
13. log 3 log 9 ( x 2 - 4 x + 3) £ 0 ; log 8 (log 1 ( x 2 - x - 6)) ³ 0
5. 6 log 3 1 - x + log 32 ( x - 1) + 5 ³ 0
2
25
lg( x 2 - 3 x + 2)
>2 ;
lg x + lg 2
18. x
P
sin
6
( x 2 - 4 x + 3) ³ -3 ;
log 2 x 3 -log 22 x -3
=
1
;
x
log
20.
log 2 ( x - 3) 2
x 2 - 4x - 5
³0 ;
21. log 21 ( x 2 + 2 x - 3) £ 1 ;
1
33
( x2 - x + ) ³ 0
6
24
2.x log 2 x + 2.x -3 log8 x - 5 = 0
log 2 ( x + 1) 2 - log 3 ( x + 2) 3
19.
>0
x 2 - 3x - 4
11. 3 log x 4 + 2 log 4 x 4 + 3 log16 x 4 ³ 0 ;
P
sin
12
;
1 log 1
( ) 2
2
2
x
£ x3
log12 (6 x 2 - 48 x + 54) £ 2
log 3 ( 3 sin 2 x - cos 2 x) £ 1
T rang 20
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