MŨ VÀ LOGARITA... PHƯƠNG TRÌNH LOGARIT... BẤT PHƯƠNG TRÌNH − HỆ PT LOGARIT.Bài 1: Giải các bất phương trình: 1/.
Trang 11) 5 .8 x−1 =500
x
x
2) 2 ( 2 4 2) (4 2 4 2)
1
−
− +
=
−
−
x
3) 1
3
−
− +
x
x
x
x
2
2 2
4) ( 5 ) ( 5 )x 1
1 -x 1
-x
+
−
≥
5) x - 1x2−4 x+3 = 1
6) ( ) ( ) 3
1 1
3
3 10 3
+
−
−
−
<
x x
x
7) 2x2−4 = 5x−2
8) 2 2 1 2
1
2
−
− ≤ x x x
9) 9x+ 9x+1 + 9x+2 < 4x+ 4x+1 + 4x+2
10) 2 1 2 3 1
1 2
1
+ + ≥ x
x
11) ( 2 1) 1 1
1
2 − + + ≥
−
x
x
x x
12) ( 2 ) 2 2 3
1 1
2
−
>
13) 7 3x+1 + 5x+3 ≤ 3x+4 + 5x+2
Ii) §Æt Èn phô:
1) 4x2−3x+2+ 4x2+6x+5= 42x2+3x+7+ 1
2) ( 7 + 4 3)sinx +( 7 − 4 3)sinx = 4 3)
( ) 2 1
12 2
1 2
.
6
23x − x − 3 x−1 + x =
4) 9x + 2 (x− 2 ) 3x + 2x− 5 = 0
5) 6 ( ) 0 , 7 7
100
72 = x+
x
x
6) 2 1 1
3
1 3
3
+
x x = 12
3
1
3
3
2 x
2
>
+
8) 9sin2x+ 9cos2x = 10
9) 4x+ 1 + 2x+ 1 = 2x+ 2 + 12
10)2 2x2 + 1 − 9.2x2 +x+ 2 2x+ 2 = 0
11)(2 + 3) (x + 7 + 4 3)(2 − 3)x = 4(2 + 3)
12) 5.32x-1-7.3x-1+ 1 - 6.3x + 9x+1 = 0
13) 6.4x - 13.6x + 6.9x =
14) 9x- 2.3x < 3 15) 4x - 6.2x+1+ 32 = 0
16)(3 5) (3 5) - 2 2 0
2 2
x -2x 1 x -2x x
-2x
≤
− +
17) 12.3x + 3.15x - 5x+1 = 20
18)32x-1= 2 + 3x-1
19) ( 6 - 35) (x+ 6 + 35)x = 12 20)
0 17 3
.
3
26
9x − x + =
21) 3 8.32x− x x+ + 4 − 9.9 x+ 4 = 0 22) 22x+1− 2x+3−64 = 0
23) ( 2 − 3) (x + 2 + 3)x = 4
24) (7 + 4 3)x − 3(2 − 3)x + 2 = 0
25) 2 1 2 1 2 1
9 6
4
2 x + + x + = x +
26) 2x2−5x+6 + 21−x2 = 2 26−5x + 1
27) 16sin2x + 16cos2x = 10
1 2
1 2
−
+
−
−
x
x x
29) 22 x+3−x−6+ 15 2 x+3−5 < 2x
30) 25 1+2x−x2 + 9 1+2x−x2 ≥ 34 5 2x−x2
31) 3log2 18. log 31 3 0
>
+
32) 32x− 8 3x+ x+4 − 9 9 x+4 > 0
33) 2log 3
1
4 9
1 3
1 − >
x− x
34) 9x− 3x+2 > 3x − 9
35) 8 3 x+4x + 91+4x > 9 x
36) 9 x2−3+1 + 3 < 28 3 x2−3−1
37) 4x2+1 32x − 4 3x + 1 ≤ 0
38)
2
5
1 2
2
log
>
x
x
39) 4x2+x+1− 2x+2+ 1 ≤ 0
1) + 2) 10) − ( ) −
Trang 23) 4 3 9 2 5 6 2
x x
x − = 4) 125x + 50x = 2 3x+1
5) 1 2 ( )2
2 -2x− x −x = x − 1 6) 2 2x + 3 3x > 6x − 1
7) - 3x 2 − 5x+ 2 + 2x > 3x.2x - 3x 2 − 5x+ 2 + ( ) 2x 2 3 x 8)
x
x
3
8
1 + 2 = ) x2 + 3 log 2x =xlog 2 5
11) − 2x −x + 2x−1 = (x− 1 ) 2 12) 3 x+4 + 2 2x+4 > 13
2 4
2 3
−
− +
−
x
14) 3x + 5x = 6x + 2
Mét sè bµi to¸n tù luyÖn:
1) 7 3x+1 - 5x+2 = 3x+4 - 5x+3
2) 6 4x - 13.6x + 6.9x = 0 3) 76-x = x + 2
4) ( 2 − 3) (x + 2 + 3)x = 4
5) 2x = 3x + 1 6) 3x+1 + 3x-2 - 3x-3 + 3x-4
= 750
7) 3 25x-2 + (3x - 10)5x-2 + 3 - x = 0
8) ( 2 + 3) (x + 2 − 3)x = 2x
9)5x + 5x +1 + 5x + 2 = 3x + 3x + 3 - 3x +1 1
2 2
x
x x
− +
=
1
14)5 5 4 0 15)6.9 13.6 6.4 0
−
17) 15 x+ = 1 4 18)2x x − +x = 4 − x
6
2
19)2 16 2
x x
− +
=
2
2
1
21)2 3 5 12 22) 1 1
x
x x x
−
28)2.16 15.4 8 0
( )
42) 2 − 5 − = 0,01 10
29) 7 4 3 x 3 2 3 x 2 0
30) 3 5 x 16 3 5 x 2x
31)3.16 2.81 5.36 32)2.4 6 9
x
+
2
34)3 4 5 35)3 4 0
x
( )
( )
2 x x
2 1
1 x
1
1 5
2 x 1
4 x 10
3 1 x-3
3x-7
1
3 39) 2 4 0,125 4 2 40) 2.0,5 -16 0 41) 8 0,25 1
x x
x x
+ + + +
+ +
−
−
÷
=
=
=
2
x
2x-1 x-1
x
43) 0,6
44) 2 -3 3 -2 45) 3.5 -2.5 0,2 46) 10 25 x 4, 25.50 x
−
=
=
x 1 x 3
x x-1
48) 4 -10.2 -24 0
=
1)
1
−
1
−
Trang 32)
+
x y
y x
3)
=
=
+
−
5
1
10
51
5 2
xy
y x x
4)
( )
= +
=
log
2 log
1 y
y
x x
5) ( )
( )
=
+
=
+
−
−
y x
x y
y
x
y
x
2
2
6 9
1 2
2
2
6)
=
=
−
12
3 3
1
log
y x
x y
7)
2
4
4
8)
( )
= +
=
−
2 log
1152
2.
3
5 x y
y
x
10)
( )
=
−
=
2 log
972 2.
3
3 x y
y x
11)
( )
+
=
−
−
−
−
=
− +
x y x
y x
y
x y
5 5
5
log
2
1
log log
12 2
log
2
48
12) (x + y )= (x2 − y2)= 3(x+ y)
3 3 3
log
=
−
+
=
−
+
0 20 2
1 log 2 log
a y
x
a y
a
14) ( )
( )
−
= +
=
y x y x
y
5
log
3
27
5 3
21) ( )
( )
= +
=
+
2 3 2 log
2 2 3
log
y x
y x y x
22)
( )
>
=
+
=
+
−
0 y 64
5, 1
5,
2 x
x x
y
y y
23)
l g
1
o x
24)
( )
=
−
=
−
1 log
1 log log
2
2
x y
x x
y
y xy
25) ( ) ( )
=
−
−
=
+
1 log log2 x 2 y
y x y
26)
( )
= +
−
=
−
9 log 2 4
36 6
2
x y
x
xx y
27) ( ) ( )
=
−
=
−
−
+ 2
1 log
log 2 2
2 2
v u
v u v
u
28)
≠
≠
=
=
0 pq
vµ q p
y
x y
x
y x
a
a a
q p
log
log log
29)
=
−
= +
5 log
log 2 2
12
1
2 x y
y x
x y
Trang 415) ( ) ( )
= +
− +
−
+
=
−
8 5
3
5 4
y x
y x y
x
y
x
xy xy
16)
>
=
=
0 x 64 2
2
2
y
y
x
x
17)
= +
= +
−
3
1 5
2
12
1 log log
2
2
5
2x y
x
y
y
x
18)
( )
>
=
+
=
+
−
0 x 8
1
10 7
2
y
x
xy y
19)
=
= +
−
32
0 5 log
2 log
2
2
xy
y
y
30)
( )
>
=
−
=
−
−
0 x 2
1
16
2 2
y x
xx y
35) ( ) ( )
o x o y
=
=
36)
( )
<
= +
=
0 a
2 2 2
2
2
lg 5, 2 lg
lg x y a
a xy
37)
=
−
=
+
1 log log
4
4 4
log log8 8
y x
y
38 ) ( )
( )
=
=
−
− +
−
−
− +
1 37
,0
1 2
16 2
8
2
2
x xy x y x
xy x y x
39)
=
−
=
+
1 log log
27 2
3 3
log log3 3
x y
y
PH¦¥NG TR×NH Vµ BÊT PH¦¥NG TR×NH LOgrIT
x
x 1
+
−
8 log 0,04 x 1 + + log x 3 1 0,2 + =
32 3 1
2
≥
33 1
3
4x 6
x + ≥ 34 log x 32( + ≥ +) 1 log x 12( − )
2
3 2
≥ + −
2
2x
log x − 5x 6 + < 1
40 log 3x x− 2(3 x − ) > 1 41
2
2 3x
x 1
5
2
+
42 x 6 2
3
x 1
x 2
+ − >÷
+
43 log x log x 022 + 2 ≤
Trang 512 3 9 x
1
2
14 ( x 1 ) ( x )
2
1
8
16.2 lg2 1( − +) lg 5( x + = 1) (lg 5 1− x + 5)
18.5 lg x = 50 x − lg5
18 lg x lg x2 2 3
19 2
log x log x
x 2 log x 1 4 x 1 log x 1 16 0 + + + + + − =
23 log x 3 5( )
8
3
5
3
5
2
+ ≥
8
2
2 log (x 2) log (x 3)
3
45 2
2
47 2
log x log x
6 + x ≤ 12 48 2 log 2x log x 2 2 3 1
x
x
− − >
49 ( x ) ( x 1 )
2
2
0
2 5x 3x
≥
51 ++ >
2 3 3
1 log x
1
1
5 log x 1 log x
2 54.2 log5x − log x 125 < 1
55 ( 1) (2 2) 3(4 ) 0
3
1 3
1 x − + log x + + log − x <
log
56.2 ( log 2 x )2 + x log 2 x ≤ 4 57 ( 2 ) ( 2 )
log x + 4x− 12 − log x + < 1 1
58 ( ) (lg 2 1)
2
1 3
lg x2− > x2 − x+
8
2
4 x + log x − 1
60 log9(3 x2 + 4 x + 2)+ 1 > log3(3 x2 + 4 x + 2)
61.logx−1( x + 1 ) > logx2−1( x + 1 )
3 2
3 3 3
2
2 x − log + x log x − log x ≥ x − + x log x
63.1 + log x 2000 < 2 64 0
1 3 2 5
5 lg
<
+
− −
+
x x x
x
65 2 4 22 ≥21
−
−
x
x logx
MỘT SỐ PHƯƠNG TRÌNH MŨ – LÔGA SIÊU VIỆT
Trang 66
/ 2
log ( 1)
log
1)2 8 14
3)log (1 ) log
4)2
5)log ( 3 ) log
−
+
=
x
x
x
x
2 5
6)log (x −2x− =3) log (x −2x−4)
2
log log 5
2
log
2
x
7) 3
8) 2.3 =3
9)log ( - 4) log 8(x+2)
10)log 3log (3 1) 1
11)3 4 0
12)3 4 5
13)3 − (3 10).3 − 3
+
+ =
+ − =
x x
x
x
x
x
2
2
x
2
x
x 6 10 2
0 14)3.4 (3 10).2 3 0
15)log log 1 1
16)4.9 12 3.16 0
17)3 os2x
=
=
x
x
x
x
c
2
1
os2x os
lg lg 6
19)9 2( 2).3 2 5 0 20)4 - 4 3.2
21)(4 15) (4 - 15) 62
23)6 12 24)6 8 10
=
x
x
2
2
25)log x−8log 2 3 x =
2 2
3 3
26) lg( 2)
8 2
27) 4 6 9
29)5 50
31)log log ( 2) 32)3log (1
=
x
x x
x
x
5
2 log ( 3)
3
4
2
) 2log 33)2 34) log (1 ) log
1 35)log ( ) log
2
+
=
=
x
x x
Trang 7MŨ VÀ LOGARIT
A PHƯƠNG TRÌNH MŨ:
Bài 1: Giải các phương trình:
1/ 3x + 5x = 6x + 2 2/ 12.9x - 35.6x + 18.4x = 0
3/ 4x = 3x + 1 4/ (3 2 2 + ) (x+ − 3 2 2)x = 6x
5/ ( 2 + 3) (x+ 2 − 3)x = 4 6/ 2x+ + 2 18 2 − x = 6
7/ 12.9x - 35.6x + 18.4x = 0 8/ 3x + 33 - x = 12
9/ 3x+ = 6 3x 10/ 2008x + 2006x = 2.2007x
11/ 125x + 50x = 23x + 1 12/ 2x2 − 1 = 5x+ 1
13/ 2x2−x− 2x+ 8 = + 8 2x x− 2 14/ 2x2 +x+ 2 2 − −x x2 = 515/
15 x2.2x + 4x + 8 = 4.x2 + x.2 x + 2x + 1 16 6x + 8 = 2x + 1 + 4.3x
17 4x2+x+ 2 1−x2 = 2 ( 1)x+ 2 + 1 18/ 3x + 1 = 10 − x.
19/ 2 2. x+ −3 x− 5.2 x+ +3 1 + 2x+4 = 0 20/ (x + 4).9 x − (x + 5).3 x + 1 = 0 21/ 4x + (x – 8)2 x + 12 – 2x = 0 22/ 3 4x = 4 3x
4x + (x − 7).2x + − 12 4x = 0 24/ 8x − 7.4x + 7.2x + 1− 8 = 0
Bài 1: Giải các phương trình:
1/ 2 3x > 3 2x 2/ ( 3 + 2) (x+ 3 − 2)x ≤ 2
3/ 2x + 2 + 5x + 1 < 2x + 5x + 2 4/ 3.4x + 1 − 35.6x + 2.9x + 1 0 5/ ( )2 ( )2 ( )
1
2x+ 1 > 2x + − 2 1 2x + + 5 6/ 4 3.2 1 1 8 0
x
+ +
−
7/ 2
9/ 2x − 1.3x + 2 > 36 10/ 2x+ + 2 11 2 − x ≥ 5
11/ 9x− 4.3x+1 + 27 0 ≤ 12/ 2x2 − − 2 3x ≤ 3x2 − − 2 3x
13/ 4x + x 1 − − 5.2x + x 1 1 − + + ≥ 16 0 14/ 32 4 0
6
x x
x + − >x
− −
15/ 6x+ < 4 2x+1 + 2.3x 16/ 1 1 2 1
2x + 2 − x 9
17/ (2 2 1x + − 9.2x + 4 ) x2 + 2x− ≥ 3 0 18/
Bài 9: Giải các hệ phương trình
1/ 2 5
y y
x
x
8
y
1
2 6
8 4
y y
x x
−
−
=
4/ 3 2 11
x
y
y x
+ = +
3 4 36
y x y x
=
3
y
Trang 87/ 2 4
4 32
x
x
y y
=
4 3 144
y x y x
− =
=
5 2 50
y x y x
=
10/ 2 3 17
3.2 2.3 6
y x
y x
+ =
x y
y x
= +
3 19
y y
x x
− =
C PHƯƠNG TRÌNH LOGARIT.
Bài 1: Giải các phương trình:
1/ log 3x+ log 9 3x = 2/ log 2 2( x− 1 log 2) 4( x+ 1 − = 2) 1
3/ 2
2 2
3 log x 9x + logx 3x = 1
5/ x.log 3 log 3 5 + 5( x − = 2) log 3 5( x+ 1 − 4) 6/ 4 log 3x +xlog 2 3 = 6
7/ ( 2 ) ( )
3 3
log x+ − (x 12) log x+ − = 11 x 0
9/ 3 log23x +xlog 3x = 6 10/ log 2 x+ = 4 log 2 2( + x− 4)
11/ 2
log x− 3.log x+ = 2 log x − 2 12/
log x.log x x+ log x+ = 3 log x+ 3log x x+
13/ 3.log 3(x+ = 2) 2.log 2(x+ 1) 14/ xlog 4 3 =x2 2 log 3x − 7.xlog 2 3
15/ 2( ) ( )
log 4x − log 2x = 5 16/ 3( 27 ) 27( 3 )
1 3 log log x + log log x =
17/ log 3x+ = − 2 4 log 3x 18/ log 2x.log 3x+ = 3 3.log 3x+ log 2x
4
2.log x= log x.log x− + 7 1 20/
log 2x− + 2 log 2x+ = 1 log 2x+ − 6
2
8 2
6.9 x+ 6.x = 13.x
log x+ log x.log x− + = 1 2 3.log x+ 2.log x− 1
24/ 3 log 2x+xlog 3 2 = 18 25/ 2
.log 2( 1).log 4 0
Bài 2: Tìm m để phương trình log 2(x− = 2) log 2( )mx có 1 nghiệm duy nhất
Bài 3: Tìm m để phương trình 2 2
log x− log x + = 3 m có nghiệm x∈ [1; 8]
Bài 4: Tìm m để phương trình log 4 2( x−m) = +x 1 có đúng 2 nghiệm phân biệt
Bài 5: Tìm m để phương trình 2
log x− (m+ 2).log x+ 3m− = 1 0 có 2 nghiệm x1, x2
sao cho x1.x2 = 27
Bài 6: Cho ph¬ng tr×nh: log32x + log23x + 1 − 2 m − 1 = 0 (2)
1) Gi¶i ph¬ng tr×nh (2) khi m = 2
Trang 9D BẤT PHƯƠNG TRÌNH − HỆ PT LOGARIT.
Bài 1: Giải các bất phương trình:
1/ log log 4( 2x)+ log log 2( 4x) ≥ 2 2/ log 2x+ ≥ 3 log 2x+ 1
log 2x − log x ≤ 1
5/ log 4 2( x − 2x 1 + )≤x 6/
log x+ 2log x− 3 x − 5x+ ≥ 4 0
2
log1 2 log
x
x x
2 2
2
2 log 2
x
≥
2
2
0 log
2
x
log log x log x 3 1
log x.log x+ ≤ 2 log x+ log x
13/ log 2 log 2 2 1
8
x x
x
÷
14/ 3 log32x+xlog 3x ≤ 6
Bài 2: Giải các hệ phương trình
1/
6
x y
+ =
2
6
y
x y
4/
6
log 3
x y
+ =
3
− =
2
log
9
y y
x x
=
log log 16
x
y
x y
y x
+ − =
10/ 22 2
log log
32 logy 4
xy x
=
12/
( )
2
2
xy
x
y
=
÷