Bài tập Bất phương trình và Logagic
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Chuyên đề phơng trình Bất ph ơng trình và Hệ phơng trình mũ Loga rit
ph ơng trình và bất ph ơng trình mũ
i) ph ơng pháp logarit hoá và đ a về cùng cơ số
1)
5008.5
1
=
x
x
x
2)
( ) ( )
244242
22
1
+=+
xxxx
x
3)
1
3
2.3
+
xx
xx
2
2
2
4)
( ) ( )
55
1x
1-x
1-x
+
+
22
5)
11-x
2
x
=
+
34x
6)
( ) ( )
3
1
1
3
310310
+
+
<+
x
x
x
x
7)
24
52
2
=
xx
8)
1
2
2
2
1
2
x
xx
9)
2121
444999
++++
++<++
xxxxxx
10)
13
12
2
1
2
1
+
+
x
x
11)
( )
112
1
1
2
+
+
x
x
xx
12)
( )
3
2
2
2
11
2
>
+
xx
xx
13)
2431
5353.7
++++
++
xxxx
Ii) Đặt ẩn phụ:
1)
1444
7325623
222
+=+
+++++
xxxxxx
2)
( ) ( )
4347347
sinsin
=++
xx
3)
( )
1
2
12
2
1
2.62
13
3
=+
xx
xx
4)
( )
05232.29
=++
xx
xx
5)
( )
77,0.6
100
7
2
+=
x
x
x
6)
1
12
3
1
3
3
1
+
+
xx
= 12
7)
12
3
1
3
3
1
x
2
x
2
>
+
+
1
8)
1099
22
cossin
=+
xx
9)
1 1 2
4 2 2 12
x x x+ + +
+ = +
10)
2 2
2 1 2 2
2 9.2 2 0
x x x x+ + +
+ =
11)
( ) ( )( ) ( )
3243234732
+=+++
xx
12)
06.3-1-7.35.3
1xx1-x1-2x
=++
+
9
13)
06.913.6-6.4
xxx
=+
14)
32.3-9
xx
<
15)
0326.2-4
1xx
=+
+
16)
( ) ( )
02-5353
2
22
x-2x1
x-2xx-2x
++
+
17)
205-3.1512.3
1xxx
=+
+
18)
323
1-x1-2x
+=
19)
( ) ( )
1235635-6
xx
=++
20)
0173.
3
26
9
=+
xx
21)
2 4 4
3 8.3 9.9 0
x x x x
+ + +
=
22)
022
64312
=
++
xx
23)
( ) ( )
43232
=++
xx
24)
( ) ( )
02323347
=++
xx
25)
111
222
964.2
+++
=+
xxx
26)
12.222
56165
22
+=+
+
xxxx
27)
101616
22
cossin
=+
xx
28)
0
12
122
1
+
x
xx
29)
xxxx
22.152
53632
<+
++
30)
222
22121
5.34925
xxxxxx
++
+
31)
03.183
1
log
log
3
2
3
>+
x
x
x
32)
09.93.83
442
>
+++
xxxx
33)
3log
2
1
1
2
4
9
1
3
1
>
xx
34)
9339
2
>
+
xxx
35)
xxxx
993.8
44
1
>+
++
36)
1313
22
3.2839
+
<+
xx
37)
013.43.4
21
2
+
+
xxx
38)
2
5
2
2
1
2
2
1
log
log
>+
x
x
x
39)
0124
21
2
+
+++
xxx
I II) ph ơng pháp hàm số:
1)
12
21025
+
=+
xxx
2)
xxx
9.36.24
=
10)
( )
0331033
232
=++
xx
xx
1
Chuyªn ®Ò ph¬ng tr×nh BÊt ph– ¬ng tr×nh vµ HÖ ph¬ng tr×nh mò Loga rit –
3)
2
6.52.93.4
x
xx
=−
4)
13
250125
+
=+
xxx
5)
( )
2
2
1
2 -2 1
x x x
x
− −
= −
6)
163.32.2
−>+
xxx
7)
( )
x
2
22
32x3x-.2x32x3x-
++−>++−
2525 xx
x
8)
x
x
381
2
=+
)
5loglog2
22
3 xx
x
=+
11)
( )
2
1
122
2
−=+−
−−
x
xxx
12)
1323
424
>+
++
xx
13)
0
24
233
2
≥
−
−+
−
x
x
x
14) 3
x
+ 5
x
= 6x + 2
Mét sè bµi to¸n tù luyÖn:
1) 7. 3
x+1
- 5
x+2
= 3
x+4
- 5
x+3
2) 6. 4
x
- 13.6
x
+ 6.9
x
= 0 3) 7
6-x
= x + 2
4)
( ) ( )
43232
=++−
xx
5)
2 3 1
x
x
= +
6) 3
x+1
+ 3
x-2
- 3
x-3
+ 3
x-4
= 750
7) 3..25
x-2
+ (3x - 10)5
x-2
+ 3 - x = 0
8)
( ) ( )
x
xx
23232
=−++
9)5
x
+ 5
x +1
+ 5
x + 2
= 3
x
+ 3
x + 3
- 3
x +1 1
( )
2
3
3 4 1
2
2
10) 1 1 11)2 4
12)8 36.3
x
x x x
x
x
x
x
−
+ − −
−
+
+ = =
=
( ) ( )
1
14)5 5 4 0 15)6.9 13.6 6.4 0
16) 5 24 5 24 10
x x x x x
x x
−
− + = − + =
+ + − =
( )
2
8 1 3
17) 15 1 4 18)2 4
x
x x x x− + −
+ = =
2
5
6
2
1 2 1 2
19)2 16 2
20)2 2 2 3 3 3
x x
x x x x x x
− +
− − − −
=
+ + = − +
( )
(
)
( )
2
2
1
1 2 2
2
4
2 2
4 8 2 5 2 6 7
21)2 .3 .5 12 22) 1 1
23) 1 24) 2 2 1
25)3 4.3 27 0 26)2 2 17 0
x
x x x
x
x
x x x x
x x
x x x x
−
− −
−
−
+ + + +
= − + =
− = − + =
− + = + − =
( ) ( )
+ + − − =
− − =
27) 2 3 2 3 4 0
28)2.16 15.4 8 0
x x
x x
( )
2 2
3
x 3 x 3 x-1
42) 2 .5 0,01. 10
− −
=
( ) ( )
+ − − + =29) 7 4 3 3 2 3 2 0
x x
( ) ( )
+
+ + − =
3
30) 3 5 16 3 5 2
x x
x
1 1 1
2 3 3
31)3.16 2.81 5.36
32)2.4 6 9
33)8 2 12 0
x x x
x x x
x
x x
+
+ =
+ =
− + =
( ) ( )
2 1 2 2 1 1 2
2
34)3 4 5 35)3 4 0
36)2 3 5 2 3 5
37) 3 2 2 1 2 0
x x x x
x x x x x x
x x
x
x x
− + + +
+ = + − =
+ + = + +
− − + − =
( )
( )
2 x
x
2 1
1 x
1
3
x
3
1
5
2 x 1
4 x 10
3 1
x-3
3
1
3x-7
1
38) 3.3 . 81
3
39) 2 4 .0,125 4 2
40) 2.0,5 -16 0
41) 8 0,25 1
x
x
x
x
x
x
+ +
+
+
+
+
−
−
=
÷
=
=
=
2
2 2 2 2
x 12 3
x
x 1 x x 1 x 2
2x-1 x-1
1 1 1
x
25 27
43) 0,6
9 125
44) 2 -3 3 -2
45) 3.5 -2.5 0,2
46) 10 25 4,25.50
x x
−
− − +
=
÷ ÷
=
=
+ =
2 2
x 1 x 3
x x-1
47) 9 -36.3 3 0
48) 4 -10.2 -24 0
− −
+ =
=
hÖ ph ¬ng tr×nh mò vµ hÖ ph ¬ng tr×nh logarit
1)
( ) ( )
2 2
log 5 log
l g l g4
1
l g l g3
x y x y
o x o
o y o
− = − +
−
= −
−
20)
( ) ( )
1
l g 3 l g 5 0
4 4 8 8 0
y
x y
x
o x o y
−
− − − =
− =
2
Chuyªn ®Ò ph¬ng tr×nh BÊt ph– ¬ng tr×nh vµ HÖ ph¬ng tr×nh mò Loga rit –
2)
( ) ( )
3 3
4 32
log 1 log
+
=
− = − +
x y
y x
x y x y
3)
=
=
+−
5
1
10515
2
xy
y
xx
4)
( )
=+
=
+
323log
2log
1
y
y
x
x
5)
( )
( )
=+
=+
−
−
yx
xy
yx
yx
2
2
69
12
2
2
6)
=
=−
12
3
3
1log
y
x
xy
7)
( )
2
4
4
9 27.3 0
1 1
l g l g lg 4
4 2
xy y
o x o y x
− =
+ = −
8)
( )
=+
=
−
2log
11522.3
5
yx
yx
10)
( )
=−
=
2log
9722.3
3
yx
yx
9)
( )
( ) ( )
2 2
l g 1 l g8
l g l g l g3
o x y o
o x y o x y o
+ = +
+ − − =
11)
( )
( ) ( ) ( )
+=−−−−
=
−+
xyxyxy
xy
555
log21
loglog122log2
483
3
12)
( ) ( )
( )
yxyxyx
+=−=+
3
22
3
33
9
logloglog
13)
( )
=−+
=−+
0202
1log2loglog
18
ayx
ayx
aa
14)
( )
( )
−=+
=+
−
yxyx
yx
xy
5
log3
27
5
3
21)
( )
( )
=+
=+
232log
223log
yx
yx
y
x
22)
( )
>=
+=
+
−
0y 64
5,1
5,2 x
xx
y
yy
23)
( )
( ) ( )
l g l g5 l g l g l g6
l g
1
l g 6 l g l g6
o x y o o x o y o
o x
o y o y o
+ − = + −
= −
+ − +
24)
( )
=−
=−
1log
1loglog
2
2
xy
x
x
y
yxy
25)
( ) ( )
=−
−=+
1loglog
22
yx
yxyx
yx
26)
( )
=+−
=
−
9log24
36
6
2
xyx
x
yx
27)
( ) ( )
=−
=−−+
2
1loglog
22
22
vu
vuvu
28)
( )
≠≠=
=
0pq vµ qp
y
x
y
x
yx
a
a
a
qp
log
log
log
29)
=
−
=+
5loglog22
12
1
2
yx
yx
x
y
3
Chuyªn ®Ò ph¬ng tr×nh BÊt ph– ¬ng tr×nh vµ HÖ ph¬ng tr×nh mò Loga rit –
15)
( ) ( )
=
+
−
+
−
+
=+
−−
8
53
542
12
yx
yx
yx
yx
xyxy
16)
( ) ( )
>=
=
0x 642
2
2
y
y
x
x
17)
=+
=+
−
3
1
52
12
1
log
log
2
2
5
2
y
x
x
y
y
x
18)
( )
>=+
=
+−
0x 8
1
107
2
yx
x
yy
19)
=
=+
−
32
05log2log2
2
1
2
xy
yx
x
y
30)
( )
>=−
=
−−
0x 2
1
16
22
yx
x
yx
35)
( ) ( )
l g l g
l g 4 l g3
3 4
4 3
o x o y
o o
x y
=
=
36)
( )
<=+
=
0a
2222
2
lg5,2lglg ayx
axy
37)
=−
=+
1loglog
4
44
loglog
88
yx
yx
xy
38 )
( )
( )
=
=
−−+
−
−−
+
137,0
12
162
8
2
2
xxyx
yx
xyx
yx
39)
=−
=+
1loglog
272
33
loglog
33
xy
yx
xy
PH¦¥NG TR×NH Vµ BÊT PH¦¥NG TR×NH LOgrIT
1.
( ) ( )
5 5 5
log x log x 6 log x 2= + − +
2.
5 25 0,2
log x log x log 3+ =
3.
( )
2
x
log 2x 5x 4 2− + =
4.
2
x 3
lg(x 2x 3) lg 0
x 1
+
+ − + =
−
5.
1
.lg(5x 4) lg x 1 2 lg0,18
2
− + + = +
6.
1 2
1
4 lgx 2 lgx
+ =
− +
7.
2 2
log x 10log x 6 0+ + =
8.
0,04 0,2
log x 1 log x 3 1
+ + + =
9.
x 16 2
3log 16 4log x 2log x− =
10.
2
2x
x
log 16 log 64 3+ =
11.
3
lg(lgx) lg(lgx 2) 0+ − =
32.
3 1
2
log log x 0
≥
÷
÷
33.
1
3
4x 6
log 0
x
+
≥
34.
( ) ( )
2 2
log x 3 1 log x 1+ ≥ + −
36.
5 x
log 3x 4.log 5 1+ >
37.
2
3
2
x 4x 3
log 0
x x 5
− +
≥
+ −
38.
1 3
2
log x log x 1+ >
39.
( )
2
2x
log x 5x 6 1− + <
40.
( )
2
3x x
log 3 x 1
−
− >
41.
2
2
3x
x 1
5
log x x 1 0
2
+
− + ≥
÷
42.
x 6 2
3
x 1
log log 0
x 2
+
−
>
÷
+
43.
2
2 2
log x log x 0+ ≤
44.
x x
2
16
1
log 2.log 2
log x 6
>
−
4
Chuyªn ®Ò ph¬ng tr×nh BÊt ph– ¬ng tr×nh vµ HÖ ph¬ng tr×nh mò Loga rit –
12.
x
3 9
1
log log x 9 2x
2
+ + =
÷
13.
( ) ( )
x x
2 2
log 4.3 6 log 9 6 1− − − =
14.
( ) ( )
x 1 x
2 2 1
2
1
log 4 4 .log 4 1 log
8
+
+ + =
15.
( )
x x
lg 6.5 25.20 x lg25+ = +
16.
( )
( ) ( )
x 1 x
2 lg2 1 lg 5 1 lg 5 5
−
− + + = +
17.
( )
x
x lg 4 5 x lg2 lg3+ − = +
18.
lgx lg5
5 50 x= −
18.
2 2
lg x lgx 3
x 1 x 1
−
− = −
19.
2
3 3
log x log x
3 x 162+ =
20.
( )
( )
2
x lg x x 6 4 lg x 2+ − − = + +
21.
( ) ( )
3 5
log x 1 log 2x 1 2+ + + =
22.
( ) ( ) ( ) ( )
2
3 3
x 2 log x 1 4 x 1 log x 1 16 0
+ + + + + − =
23.
( )
5
log x 3
2 x
+
=
24.
( )
2
8
log x 4x 3 1− + ≤
25.
3 3
log x log x 3 0− − <
26.
( )
2
1 4
3
log log x 5 0
− >
27.
( )
( )
2
1 5
5
log x 6x 8 2 log x 4 0
− + + − <
28.
1 x
3
5
log x log 3
2
+ ≥
29.
( )
x
x 9
log log 3 9 1
− <
30.
x 2x 2
log 2.log 2.log 4x 1>
31.
8 1
8
2
2log (x 2) log (x 3)
3
− + − >
45.
2
3 3 3
log x 4log x 9 2log x 3− + ≥ −
46.
( )
2 4
1 2 16
2
log x 4log x 2 4 log x+ < −
47.
2
6 6
log x log x
6 x 12+ ≤
48.
3
2 2
2 log 2x log x
1
x
x
− −
>
49.
( ) ( )
x x 1
2 1
2
log 2 1 .log 2 2 2
+
− − > −
50.
( ) ( )
2 3
2 2
5 11
2
log x 4x 11 log x 4x 11
0
2 5x 3x
− − − − −
≥
− −
51.
+
>
+
2
3
3
1 log x
1
1 log x
52.
+ <
− +
5 5
1 2
1
5 log x 1 log x
53.
− >
x 100
1
log 100 log x 0
2
54.
11252
5
<−
x
logxlog
55.
( ) ( ) ( )
04221
3
3
1
3
1
<−+++−
xlogxlogxlog
56.
( )
xlogxlog
x
2
2
2
2
+
≤ 4 57.
( ) ( )
2 2
5 5
log 4 12 log 1 1x x x+ − − + <
58.
( ) ( )
12lg
2
1
3lg
22
+−>− xxx
59.
( )
3
8
2
4
1
−+
xlogxlog
≤ 1
60.
( ) ( )
2431243
2
3
2
9
++>+++
xxlogxxlog
61.
( ) ( )
11
1
1
2
+>+
−
−
xlogxlog
x
x
62.
( )
( )
2
3
23
33
2
3
43282 xlogxxxlogxlogxlogx
+−≥−+−
63.
220001
<+
x
log
64.
0
132
5
5
lg
<
+−
−
+
x
x
x
x
65.
2
1
2
24
2
≥
−
−
x
x
log
x
MỘT SỐ PHƯƠNG TRÌNH MŨ – LÔGA SIÊU VIỆT
5
