Tổng hợp kiến thức Phương trình - Bất phương trình - Hệ phương trình mũ - Hệ bất phương trình mũ

 ) Û 1133
log23
log3log =++ xxx Û 11log3
1
log
2
1
log 333 =++ xxx 
Ûlog3x = 6 Û x = 3
6 = 729. V Ëy ph-¬ng tr×nh ®· cho cã nghiÖm lµ x = 729. 
B µi tËp t-¬ng tù 
1. 1)(loglog 23 =x 
2. 0)3(log 222 =-+ xx 
3. log2(x
2 – 4x – 5) £ 4 
4. log12(6x
2 – 4x – 54) £ 2 
5. ( ) 05loglog 24
2
1 >-x 
6. log3(5x
2 + 6x + 1) £ 0. 
7. ( ) 441log 2
2
1 £--+ xx 
8. ( ) 012log 2
5
1 £+-- xx 
9. 1
3
1
9log 23 -³÷
ø
ö
ç
è
æ +-- xx 
10. log2(25
x + 3 –1) = 2 + log2(5
x + 3 + 1) 
11. logx(2x
2 – 7x +12) = 2 
12. log3(4.3
x – 1) = 2x – 1 
13. log2(9 - 2
x) = 3 – x 
14. log2x – 3 16 = 2 log2x – 3x = 2 
15. 1
1
32
log3 <-
-
x
x
 (SPVinh’98) 
16. 
5
1
log25log2 5 x=- 
17. 02log
3
1
log 3
5
1 =÷
ø
ö
ç
è
æ -x 1
11
1
log2 =÷
÷
ø
ö
ç
ç
è
æ
--x
18. 01,02log 10 -=x 2)1352(log
2
7 =+- xx 
19. 3logloglog
2
142 =+ xx log2(|x+1| - 2) = - 2 
20. log2(4.3
x - 6) - log2(9
x - 6) = 1 3)62(log2 =+xx 
21. ( ) xx 323 log21log =++ 3log3x – log9x = 5 
22. log(2(x – 1) + log2x = 1 logx + 1(3x
2 – 3x – 1) = 1 
23. 
x
x
x
x
-
=
+
2
log
1
log 33 ; 1
log)1(log 55 +
=-
x
x
x 
24. )1lg(
2
1
lg += xx ; 
2
1
2
12
log4 -<+
-
x
x
 (§ HVH’98) 
25. 
( )
3
40lg
11lg
3
=
-
++
x
x
 ; ( )
1
log1log 55 +
=-
x
x
x 
26. ( )( ) 5lg2lg210lg 21lg 2 -=-- xx 
27. ( ) 1log296log 32
2
8 -+- = xxx x )22(
3
1
)43(
3
1 loglog
2 +-+ = xxx 
28. log4(log2x) + log2(log4x) = 2 
29. logx + 1(2x
3 + 2x2 – 3x + 1) = 3 
30. log2x.log3x = log2x
2 + log3x
3
 – 6 
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò 
 T rang 12 
31. 0
63
2
log
21
32
log
7
17 =-
+
+
x
x
32. ( ) 0
2
6
log1log
3
13 =-
+-
x
x 
33. log3x.log9x.log27x.log81x = 
3
2
34. 0
4
2log
2
1log 2
2
1 =-+÷
ø
ö
ç
è
æ -
xx
35. ( )42221 21log
3
4
1
log 2
x
x
x
-
-=
- 
36. 
8
3
log33log31log 222 -+=x 
37. 2
1
18
log
2
2 £+
-+
x
xx
 (QGHN’99) 
38. 3logloglog
2
142 =+ xx 
39. 
2
11
logloglog 842 =++ xxx 
40. ÷
ø
ö
ç
è
æ=++
12
11
3
5
log3loglog 2793 xxx 
41. log2(x + 3) + log2(x – 1) = log25 
42. ( )
x
xx
4
4 log
2
10log.2log21 =-+ 
43. ÷
÷
ø
ö
ç
ç
è
æ
-=+÷
ø
ö
ç
è
æ + x
x
1
327lg2lg3lg
2
1
1 
44. ( )12log
1
2
3
2
log 2
3
2
12 2 -
-=÷
ø
ö
ç
è
æ -
- x
x
x
45. 2)23lg()32lg( 22 =--- xx 
46. 2lg)65lg()1lg(lg --=-+ xxx 
47. 7logloglog 2164 =++ xxx 
48. 1+lg(1+x2 – 2x) – lg(1 + x2) = 2lg(1 – x) 
49. 2 + lg(1 + 4x2 – 4x) – lg(19 + x2) = 2lg(1 – 2x) 
50. ( ) 2lg
2
5
lg1lg
2
1
lg2 +÷
ø
ö
ç
è
æ +=--÷
ø
ö
ç
è
æ + xxx 
51. ( ) xxxx lg
2
1
6lg
2
1
3
1
lg
3
4
lg -+=÷
ø
ö
ç
è
æ --÷
ø
ö
ç
è
æ + 
52. (x – 4)2log4(x – 1) – 2log4(x – 1)
2 = (x – 4)2logx-1 4 – 
2logx - 116 
53. 32
)123(
2
)23(
2 log3loglog
22
+=+ ++++ xxxx 
54. ( ) 944log2log 2323 =++++ xxx 
55. 
2
11
loglog3log 3
12525
3
5 =++ xxx 
56. 0log
2
log 1 =-
-
x
a
xa
a
a
57. 6logloglog
3
133
=++ xxx 2log2log.2log 42 xxx = 
58. log2x + log4x + log8x = 11. log2x – log16x = 3 
59. 3log )34(
2
=-+ xxx ; 1)(loglog
2
1
3
1 -=
x 
60. lg5 + lg(x + 10) = 1 – lg(2x – 1) + lg(21x – 20) 
61. x
x
x
x 2
3
323 log2
1
3
loglog.
3
log +=-÷
ø
ö
ç
è
æ 
62. 1)2(loglog 33 =++ xx ; x(lg5 – 1) = lg(2x + 1) – lg6 
63. 2loglogloglog 4224 =+
xx ; 6lg5lg)21lg( +=++ xx x 
64. ÷
ø
ö
ç
è
æ -=+
2
11
4
75
log
2
log
1
3
2
32
x
x x 
65. 0)2(loglog 2
322862 22
=-
++++
xx
xxxx
; 
66. xxxx 10
)1(
432 loglogloglog =++
+ 
67. 3
2log
1
16
32
log
56
2 -=÷
ø
ö
ç
è
æ -
x
x
xx
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò 
 T rang 13 
68. ( ) 13log25log
3
1
82 =-+- xx 
69. log4log3log2x = 0 logplog2log7x £ 0 
70. )12(log.3log21
log
2log21
9
9
9 x
x x
-=-
+
71. [ ]{ }
2
1
log31log2log 234 =+ x 
72. )93.11(5
)33(
5
3
5 logloglog)1(
1 -+ =+-
+ xx
x 
73. )1(log)1(log)1(log 543 +=+++ xxx 
74. [ ]{ }
2
1
log1(log1log1log =+++ xdcba 
75. ( )[ ]{ }
2
1
log31log1log2log 2234 =++ x 
76. lg5 + lg(x + 10) = 1 – lg(2x – 1) + lg(21x – 20) 
77. log2(x
2 + 3x + 2) + log2(x
2 + 7x + 12) = 3+ log23 
78. 2log3(x – 2)
2 + (x – 5)2logx – 23 = 2logx – 29 + 
(x – 5)2log3(x – 2) 
79. ( ) 3log
2
1
log.265log 33
122
9 -+
-
=+- - x
x
xx 
80. 0logloglog 5
3
12 >x ; 2
1
logloglog
524
=x 
v Ph-¬ng ph¸p ®Æt Èn sè phô 
L o¹i 1: 
V Ý dô. Gi¶i ph-¬ng tr×nh: 1
lg1
2
lg5
1
=
+
+
- xx
Gi¶i: 
§ Ó ph-¬ng tr×nh cã nghÜa, ta ph¶i cã: lgx ¹ 5 vµ lgx ¹ -1.§ Æt lgx = t (* ) (t ¹ 5 , t ¹ 1), ta ®-îc pt Èn t: 
( )
ê
ê
ê
ê
ë
é
=
+
=
=
-
=
Û=+-Û--+=-++Û
+-=-++Û=
+
+
-
=--=D
3
2
15
2
2
15
065552101
)1)(5()5(211
1
2
5
1
16.4)5(
22
2
t
t
ttttttt
tttt
tt
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 = 102 = 100. 
+ V íi t = 3, thay vµo (* ) ta cã: lgx = 3 Û x = 103 =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. 01log2log.4 24
2
4 =++ xx 
2. 5log25,155log 2xx =- 
3. 0log610log10log 1023 =-+ xxx 
4. 1)15(log).15(log 242 =--
xx 
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò 
 T rang 14 
5. 12log.)2(log 222 =xx 
6. 02log.4)33(log
332
=-+
+x
x 
7. 34log2log 22 =+ x
x
8. 9)(lglg3 22 =-- xx 
9. 40lg9lg 22 =+ xx 
10. 03log3log.3log
813
=+ xxx 
11. ( )
2
5
3log14log
143
>++
+x
x 
12. 2
7
1
loglog7 =- xx 
13. log2x + logx2 = 
2
5
14. logx2 – log4x + 
26
7
 = 0 
15. 044loglog.5 22 =-- xx 
16. ( ) 025log
2
1
log 5
2
5 =-+ xx 
17. 316log64log 22 =+ xx 
18. 
2
2
2
2 log23log
xx ³+ 
19. 2log3(2x + 1) = 2.log2x + 13 + 1 
20. 12log.)2(log 222 <xx 
21. 
1lg
2
lg
1lg
lg2
-
+-=
- x
x
x
x
22. 022.64 27logloglog 399 =+- xx 
23. 022.54 9logloglog 333 =+- xx 
24. xx x 2
2
2 log21log2 =+ - 48 
25. xx x 2
2
2 log21log 2242 =++ 
26. ( ) ( ) 52log82log
4
1
2
2 ³--- xx 
27. 1log)1(log).1(log 26
2
3
2
2 --=-+-- xxxxxx 
28. 09log42log 24 =++ xx log2|x + 1| - logx + 164 = 1 
29. ( ) ( ) 2422 116log16log2
2
3
2
3 =+ +-- xx 013loglog3 33 =-- xx 
30. ( ) xx ++ =- 2log2log2 55 525 ; 011
4
log.3log.2 2
2
2 =--
x
x 
31. 1loglog2 1255 <- x
x ; 01lg10lg 322 >+- xx 
32. 04log)1(log )1(2
32
2 <--+
+xx ; 9)(lglg3 222 =-- xx 
33. 3lg)1,0lg().10lg( 3 -= xxx ; 01log2)(log4
2
4
2
4 <++-
xx 
34. aaa xx
3
3 logloglog =- ; 8lg3x – 9lg2x + lgx = 0 
35. 1log32log
3
1
3
1 +=+- xx ; 
36. ( ) ( )
a
axax
axa
1
loglog.log 2= 
37. ( ) xx x 27log27 log3
10
log1 27 =+ 
38. lg4(x – 1)2 + lg2(x – 1)3 = 25 (Y HN’00) 
39. 2log4(3x – 2) + 2.log3x – 24 = 5 
40. 2log8loglog.5 29
3
9
9
2 =++ xxx x
x
x 
41. ( ) ( )243log1243log 2329 +->++- xxxx (SPHN’00) 
42. ( ) ( ) 022log32log 2
2
1
22
2 £++-++- xxxx 
43. 05log4log 42 =-- xx (C§SPHN’97) 
44. log3x + 7(9 + 12x + 4x
2) + log2x + 3(21 + 23x + 6x
2) = 4 
45. log1-2x(1 - 5x + 6x
2) + log1 - 3x(1 - 4x + 4x
2) = 2 
46. 1log)1(log).1(log 26
2
3
2
2 --=-+-- xxxxxx 
L o¹i 2: § «i khi ®Æt Èn phô nh-ng ph-¬ng tr×nh vÉn chøa Èn ban ®Çu. 
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò 
 T rang 15 
1. 0log24log.lglg 22
2 =+- xxxx 
2. 016log)5()1(log )1(5
2
5 =--++
+xxx 
3. 03log)4(log 2
2
2 =-+-+ xxx
x 
4. 062log).5(log 2
2
2 =+--+ xxxx 
5. 016)1(log)1(4)1(log)2( 3
2
3 =-+++++ xxxx 
6. 016)2(log)2(4)2(log)3( 3
2
3 =-+++++ xxxx 
7. log2
2x + (x – 1)log2x + 2x – 6 = 0. (TS’97) 
L o¹i 3: 
1. 0log.loglogloglog 3232
2
2 =-+-
xxxxx 2. 2log.loglog)(log2 )(222
2
2
2
=-+- -xxxxxx 
L o¹i 4: 
 4.1) 2)1log(3)1(log 222 =-++-- xxxx 4.6) 3 3
3
2 2log332log -=+
xx 
4.2) 1lg1lg23 --=- xx 4.7) 11loglog 2
2
2 =++ xx 
4.3) 1log1log1 3 33 3 =++-
xx 4.8) 12log36 )15(6 ++=
+ xxx 
4.4) 6log52log3 )4(4
)4(
4
22
=-++ -- xxxx 4.9) 1log.67 )56(7
1 += -- xx 
4.5) 10lg1 2 =-+ xx 
v Ph-¬ng ph¸p sö dông tÝnh ®¬n ®iÖu cña hµm sè l«garit 
L o¹i 1: 
log2(3x – 1) = -x + 1 2. 4log
3
1 -= xx 3. 4log3 =+
xx 4. 5log2
2
1 =+ xx 
L o¹i 2: Ph-¬ng tr×nh kh«ng cïng c¬ sè. 
V D 1. Gi¶i ph-¬ng tr×nh: 
1. x
x
=
+ )3(
5log2 
2. xx =+ )3(log52 
3. x
x
=
- )3(
2log3 
4. xx 73 loglog £ 
5. xx 25
)1(
2 loglog ³
+ 
6. ( ) xx 32 log1log =+ 
7. ( ) xx 32 log1log =+ 
8. )22(2
2
3 log1log
-->+ xx 
9. xxx 4
84
6 log)(log2 =+ 
10. ( ) xxx 342 log4
1
log =+ 
11. xx coslogcotlog.2 23 =
L o¹i 3: f(x) = f(y) Û x = y, (f - ®ång biÕn hoÆc nghÞch biÕn ) 
1. log3(x
2 + x + 1) – log3x = 2x – x
2. 2. 
x
xxx -=- -
1
log22 2
1 
1. T×m k ®Ó ph-¬ng tr×nh cã ®óng 3 nghiÖm: ( ) ( ) 022log.232log.4
2
1
22
2
2
=+-++- +--- kxxx xxkx 
T×m m ®Ó ph-¬ng tr×nh cã nghiÖm. 1) axx =++- )54(log
23
 ; 2) axx =-+ )2(log 442 
L Ëp b¶ng xÐt dÊu: 
Ph-¬ng tr×nh, bÊt ph-¬ng tr×nh Mò 
 T rang 16 
1. 
( ) ( )
0
43
1log1log
2
3
3
2
2 >
+-
+-+
xx
xx
2. 
( ) ( )
0
43
1log1log
2
3
3
2
2 >
-+
+-+
xx
xx
3. 
( ) ( )
0
43
1log1log
2
3
2
2 >
--
+-+
xx
xx
4. 
( ) ( )
0
43
1log1log
2
3
32 >
-+
+-+
xx
xx
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 pt sau cã 2 nghiÖm tm x 1, x2 tm: 4 < x1 < x2 < 6: 
( ) ( ) ( ) 024log)12(4log3
2
1
2
2
1 =++-+--- mxmxm 
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: 
(a – 4)log2
2(2 – x) – (2a – 1)log2(2 – x) + a + 1 = 0 
4. T×m c¸ c gi ¸trÞ cña m ®Ó ph-¬ng tr×nh sau cã hai nghiÖm x1, x2 tho¶ m·n x 1
2 + x2
2 > 1: 
2log[2x2 – x + 2m(1 – 2m)] + log1/2(x
2 + mx – 2m 2) = 0 
5. (§HKT HN ’98) Cho ph-¬ng tr×nh: ( ) 3)2(4log )2(22 2 -=- - xx x a 
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 biÖt x 1, x2 tho¶ m·n: 4,
2
5
21 ££ xx . 
6. V íi gi ¸trÞ nµo cña a th× ph-¬ng tr×nh sau cã