BÀI T P CH NG II ĐI S 12Ậ ƯƠ Ạ Ố
PH NG TRÌNH VÀ B T PH NG TRÌNH MŨ – LÔGARITƯƠ Ấ ƯƠ
Lũy thừa
( )
( )
. 0
1 1
. . 1
n
m
m n m n m m m m n n n
n n
n n n m m
m n
n nm n m m
n n
n n
a a a a b ab a a a a a
a a
a b a a a
a a a a a
b a a b b
+ −
−
−
−
= = = = = =
� � � � � �
= = = = =
� � � � � �
� � � � � �
Lôgarit
( )
log 0,1 0
a
a b b a b
α
α
= = > >� �
Cho
1 2
, , , , 0, , , 1a b c b b a b c
>
( )
log 1
1 2 1 2 1 2
2
1
log 1 0 log 1 log log log log log log
log . log log log log log
log 1
log log log log log
log log
n n
a
m m
a a a a a
aa a
b
a a a a a a
c
a a c c
a
c b
m
a b m b b b b b
n n
b
a b b b b b b b
b
b
b b a b b
a a
= = = = =
� �
= = + = −
� �
� �
= = =
Chú ý:
2
2 log log
n
a a
n x x
=
còn
2
log 2 log
n
a a
x n x
=
Các công thức đạo hàm:
( )
1
'x x
α α
α
−
=
( )
sin ' cosx x
=
( )
1
ln 'xx
=
'
2
1 1
x x
� �
= −
� �
� �
( )
2
1
cot ' sin
xx
= −
( )
1
log ' ln
a
xx a
=
( )
'
1
2
xx
=
( )
'
x x
e e
=
( )
' ln
x x
a a a=
( )
' ' 'u v u v
=
( )
. ' '. . 'u v u v u v
= +
'
2
'. . 'u u v u v
vv
−
� �
=
� �
� �
I. ĐO HÀMẠ
Tìm đo hàm c a các hàm s sau:ạ ủ ố
1/
( )
1
x
y x e
= −
2/
( )
1
2
x x
y e e
−
= −
3/
( )
ln sin 2y x
=
4/
( )
2
3 2 lny x x
= −
5/
2
ln x
yx
=
6/
x
e
yx
=
7/
sin
x
y e x
=
8/
sin x
y e=
II. TÌM GTLN – GTNN
Bài 1: Tìm GTLN – GTNN c a các hàm s sau: ủ ố
a/
( )
3 2
2 3 12 2f x x x x
= − − +
trên
[ ]
2;0−
b/
( )
2x x
f x e e
−
= +
trên
[ ]
1;2−
c/
( )
2
2lnf x x x
= −
trên
[ ]
2;2−
Bài 2: Tìm GTLN – GTNN c a các hàm s sauủ ố
a/
( )
ln 2 2f x x x x
= − +
trên
2
1;e
� �
� �
b/
( )
2
5f x x
= −
c/
( )
( )
2
1
x
f x e x x
= − −
trên
[ ]
0;2
d/
9
( )f x x x
= +
trên đo n ạ
[ ]
2;4
III. PH NG TRÌNH MŨ:ƯƠ
Bài 1: Ph ng trình c b nươ ơ ả
1/
3 27
x
=
2/
1
1
5125
x
−
=
3/
6 2
x
= −
4/
3
6 5
x
+
=
5/
1 1
2.3 6.3 3 9
x x x
+ −
− − =
6/
1 1
5 10 .2 .5
x x x x
− − +
=
7/ 3x + 3x+1 + 3 x+2 = 351
8/
3
2 log
3 81
x
−
=
9/
2 1
5 7
x x
−
=
Bài 2: D ng đt n phạ ặ ẩ ụ
1/
1
4 3.2 16 0
x x
−
+ − =
2/
2
3 5.3 6 0
− + =
x x
3/
3.4 21.2 24 0
− − =
x x
4/
16 17.4 16 0
− + =
x x
5/
1 1
3 3 10
x x
+ −
+ =
6/
2 1
5.4 2.6 3
x x x
+
− =
7/
4.9 12 3.16 0
x x x
+ − =
8/ / 4x – 6.2x+1 + 32 = 0 9/
( ) ( )
2 1 2 2 1 1
x x
+ − − =
Bài 3: T ng h pổ ợ
1/
1
5 .8 500
x
xx
−
=
2/
2 3
3.2 2 2 60
+ +
+ + =
x x x
3/ 4x + 10x = 2.25x
4/
2 1
3 .5 7 245
− −
=
x x x
5/
12
3 6
3 3 80 0
−
− − =
x x
6/
1 2 1 2
5 5 5 3 3 3
x x x x x x
+ + + +
+ + = + +
7/
1
3 18.3 29
+ −
+ =
x x
8/
3
2 log
3 81
−
=
x
x
9/
6 3
3. 2 0
− + =
x x
e e
10/
( )
2 2
1
2 9 3 2 2
x x x x
+
+ = +
11/
5 3
3 5
x x
=
12/
( ) ( )
7 4 3 2 3 6
x x
+ + + =
13/
3 5 2
x
x
= −
IV. PH NG TRÌNH LÔGARIT:ƯƠ
1/
2
log 3x=
2/
( )
2
1
2
log 1 9x− =
3/ log3(x + 1) + log3(x + 3) = 1
4/
2
ln 3ln 2 0x x− + =
5/
2 4
log log ( 3) 2
− − =
x x
6/ log(x – 1) – log(x2 – 4x + 3) = 1
7/
1
2 2
log (2 1).log (2 2) 6
+
+ + =
x x
8/
2
6log 1 log 2
= +
x
x
9/
3 9 1
3
log log log 2x x− =
10/
( ) ( )
2
2 2
log 2 5log 2 6x x− + − =
11/
( ) ( )
1
5 25
log 5 1 .log 5 5 1
x x+
− − =
12/
( )
2
log 3.2 1 2 1
x
x− = +
13/
( )
3 3
log log 2 1x x− − =
14/
( ) ( )
3 3
log 2 1 log 1x x− = −
15/
2
2 2 2
log ( 8) log log 6x x
+ = +
16/
2
2 4
2log 14 log 3 0x x− + =
17/
2
4 2
log log 5x
=
18/
( )
3
log 3 8 2
x
x+ = +
19/
( )
2 2
log log 1 1x x+ − =
20/
( )
4
log 2 .log 2 1
x
x
+ =
21/
( )
1 2
log 4 1 log 1
x
x
−
= + −
22/
2 3
log 20log 1 0x x− + =
23 /
9
4log log 3 3
+ =
x
x
24/
2 4 16
log log log 7
+ + =
x x x
25/ log
3
( )
3 1
−
x
.log
3
( )
1
3 3
+
−
x
= 6.
26/ lg2x – lg3x + 2 = 0 27/
2 2
log ( - 3) +log ( - 1) = 3x x
V. B T PH NG TRÌNH MŨ – LÔGARIT:Ấ ƯƠ
1/
1
2 4
x
+
2/
1
1
24
x
+
>
3/
2 1
3 3
4 4
x+
4/
2
2 3
3 4
4 3
−
� �
� �
� �
x x
5/
1 1
2.3 6.3 3 9
x x x
+ −
− − <
6/
3
log 1x>
7/
0,3
log 1x
8/
5 5
log log (2 3)x x
> −
9/
1 1
2 2
log log (3 )x x
−
10/
2 2
2 2
log 5 3log
+
x x
11/
2.9 4.3 2 1
+ + >
x x
12/
0,5
2 1 2
5
log
+
+
x
x
13/ log
3
( )
2+x
log
9
( )
2+x
14/ 9x - 4.3x +3 < 0
15/
ln 5x
>
16/
2 2
log ( - 3) +log ( - 1) < 3x x
17/
2
lg 3lg 2 0x x− +