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kIẾN THỨC CƠ BẢN VỀ hàm số lôgarit

1 và N > 0

log a N

a

0

a

1

N


( a > 0, a

*a>1
:
*0

1)

0


y
y

O

0
a>1

à

à

x

1

x

1

O

y=logax

y=loga x

garit:
ln x '
ln u '

1
x
u'
u




ln x '



ln u '

1
x
u'
u

à


log a x '

log a u '

a/ y

log 1
2

2

x 1
0
x 1

x 1
0
x 1
1;

log 1 log 5
5





log a u '

log a x '

1
x ln a

u'
u.ln a

à

à

x 1
.
x 5

log 1

b/ y

u'
u.ln a

1
x ln a

x2 1
.
x 3

x
x
x
x

1
1
1
1
0
1

x 1
1 0
x 1
x
1 x 1

2
0 x
x 1
x
1 x 1

1


x2 1
x 3

log 1 log 5
3

x2

2

x 1
1
x 3
x2 1
0
5
x 3

2

x 1
1
x 3
x2 1
5
x 3

0 log 5
0

3

x
x

1 x 2
3 2 x 7

x 2
0
x 3
x 2 5 x 14
0
x 3
x
3

0

x

3; 2

2; 7

Bài 2. Tính giá

a.

31

log3 4

5

1
2 .3 log5 2
3

7 log7 4

3
4

72 7 log7 9

2 log 7 6

4 4 19

b.

1

c. 72 49 2
1

72 49 2

log7 9 log7 6

log7 9 log7 6

5

5

log

log

5

4

3

4

5

2 log 5 4

72

9
36

1
16

d.

-GA-RÍT
Bài 1
a. A log 9 15 log 9 18 log 9 10

18 4,5

22,5


A

log 9 15 log 9 18 log 9 10

3

1
log 1 400 3log 1 3 45
2
3
3

2 log 1 6
3

1
log 1 3
2
6

log 36 2

1
log 3 3 3
2

log 9 3 3

3
2

36.45
20

log 1
3

log 1 9 2

log 3 3 4

4

3

1
log 1 3
2
6

c. C log 36 2

C

15.18
10

1
log 1 400 3log 1 3 45
2
3
3

b. B 2 log 1 6

B

log 9

1
1
log 6 2
log 6 3
2
2

1
log 6 2.3
2

1
2

d. D log 1 log 3 4.log 2 3
4

D log 1 log 3 4.log 2 3

log 4 log 2 3.log 3 4

1
log 2 2
2

log 4 log 2 4

4

1
2

Bài 2. Hãy tính
a. A log 2 2 sin

A log 2 2 sin

b. B log 4
B

log 4

3

3

7

log 2 cos

12
3

7

3

log 2 cos

12

3

3

3

log 2 2 sin

12
3

49

3

21

49

3

21

3

log 4

log 4

12

3

12

.cos

log 2 sin

12

log10 tan 4 log10 cot 4

d. D log 4 x

D

9

log tan 4.cot 4

log 4

3

7

log1 0

1
log 4 216 2 log 4 10 4 log 4 3
3

1
log 4 216 2log 4 10 4log 4 3
3
1
6.34
log 4 63 log 4 10 2 log 4 34 log 4 2
3
10

6

1
2

1

9

c. C log10 tan 4 log10 cot 4
C

log 2

log 4 x

x

35
50

3

3

3

49

3

21

3

9

log 4 7 3

1


Bài 3. Hãy tính :
a. A log a a 3 a 5 a
A log a a 3 a 5 a

log a a

3

1 1
2 5

1
2

3

1
5

37
10

1

27
10

b. B log a a 3 a2 5 a a
1

1

log a a 3 a 2 5 a a

B

c. log 1
a

1

log a

4

a a

a

3

1
3

1

3

3
10

a 5 a3 3 a 2
a4 a

a 5 a3 3 a2

log 1

log a a

1 1
2
2 5

a

a

3 2
5 3
1 1
2 4

34
15

3
4

91
60

d. log tan10 log tan 20 log tan 30 .... log tan 890
log tan10 log tan 20

( vì : tan 890 cot10

log tan 30 .... log tan 890
tan10 tan 89 0

log tan10 tan 890.tan 20.tan 87 0...tan 450

tan10 cot10 1

0

)

e. A log 3 2.log 4 3.log 5 4......log15 14.log16 15
A

log 3 2.log 4 3.log 5 4......log 15 14.log16 15 log16 15.log15 14....log5 4.log4 3.log3 2

f. A

A

1
log 2 x
1
log 2 x

log x 2011!

Bài 4

1
log 3 x
1
log 3 x

1
1
..........
log 4 x
log 2011 x
1
1
..........
log 4 x
log 2011 x

x

log16 2

1
4

2011!

logx 2 logx 3 ... logx 2011 logx 1.2.3...2011

log 2011! 2011!

1


a 2 b2

a.

c2; a

a2

2

1
log c b a

0, c 0, c b 1 , thì :
log c b a log c b a 2 log c b a.log c b a

0, b

c2 b2

1
log c b a

c b c b

2

2 log c b a.log c b a

log a c b

log a c b

log c b a log c b a

1
log a N
log c N

log a N log b N
a, b, c 1
log b N log c N
b2

1
log b N
log b N log c N
log c N .log b N

2 log N b log N a log N c
log a N log b N
log a N .log b N

1
log a N
log a N
log c N

ac

1
1
log c N log b N
log a N log b N
log b N log c N

log x a, log y b, log z c
log b y

2 log a x.log c z
0
log a x log c z

x, y, z , a, b, c 1

log x a, log y b, log z c
1
log a x

1
log c z

log x a log z c 2log y b

2
log b y

log b y

2 log a x.log c z
log a x log c z
a2 b2

a 2 b2

2 ln

a b
3

7 ab

a b

ln a ln b

ln

2

9ab

a b
3

a b
3
ln a ln b
2

7 ab

2

ab .

e
log ax bx

log ax bx

f

log a b log a x
1 log a x
log a bx
log a ax

log a b log a x
1 log a x

ln

VP

dpcm

hai

a b
3

ln a ln b
2


1
log a x

1
1
.........
log a 2 x
log a k x

VT= log x a log x a 2 ...log x a k

k k 1
2 log a x

1 2 3 ... k log x a

k 1 k
2 log a x

VP

Bài 1. Tính
a. A log 6 16
A

log12 27

log 6 16

x

log 3 27
log 3 12

log 3 2 4
log 3 6

4 log 3 2
1 log 3 2

log 2 5

a; log 2 3 b

log12 27

A log 6 16

b. C log 3 135

log 3 135 log 3 5.33

C

c. D log 6 35

x

log 3 5 3

log 27 5

a; log 8 7

3
1 log 3 4

x

log 2 5
3
log 2 3
b; log 2 3

log 3 4

a
3
b

3
1
x

3 x
3 x
(*)
log 3 2
x
2x
2 3 x .2 x 12 4 x
x x 3
x 3

a 3b
b

c

1
1
log 3 5 log 3 5 3a; b log 8 7
log 2 7 log 2 7 3b (*)
3
3
log 2 5.7 log 2 5 log 2 7 log 2 3.log 3 5 log 2 7 b.3a 3b
log 6 35
log 2 2.3
1 log 2 3
1 log 2 3
1 b

Ta có : a log 27 5
Suy ra : D

d. Tính : log 49 32
Ta có : log 2 14 a

log 2 14
1 log 2 7
5

log 49 32

log 2 2
log 2 7 2

5
2 log 2 7

a

a
log 2 7

a 1

5
2 a 1

Bài 2
a. A

A

log a b log b a 2 log a b log ab b log b a 1

log a b log b a 2 log a b log ab b log b a 1

log a b 1
log a b

2

1 log ab a

1

3b a 1
b 1


2

log a b 1
log a a
log a b 1
1
1
log a b
log a ab
log a b
log a b 1
1
1
log b a
log a b
log a b

b. B log 2 2 x 2
log 2 2 x 2

B

1 3log 2 x

c. C

C

log 2 x x

log 2 x x

log 2 x

2

1
1
1 log a b

1

2

log a b
1 log a b

1
log 22 x 4 1 2 log 2 x log 2 x log 2 x 1
2
2
9 log 2 x
3log 2 x 1

log x log 2 x 1

2

log a p log p a 2 log a p log ap p

log a p log p a 2 log a p log ap p

log a p 1
log 2a p
log a p
1 log a p

log a b 1
log a b

1
4 log 2 x
2

log a p

log a p 1
2
a

log p

2

log a p

log a p
1 log a p

log a p

log a b 3; log a c

a. x a 3b 2 c
Ta có : log a x log a a 3b 2 c

c. x

3 2 log a b

1
log a c 3 2.3 1 8
2

a4 3 b
c3

4

1
log a c 3log a c
3

4

1
3

2

6 10

a 2 4 bc 2
3
ab 4 c

Ta có :
log a x

23

a4 3 b
c3

a 2 4 bc 2

log a
2

Bài 4

3

3
4

ab
4

4

c

log a p

3

log a p

log a x

Ta có : log a x log a

2

log a p

Bài 3

b. x

1

1
log 22 x 4
2

log x log 2 x 1

8 log 2 x

2

2

1
log a b 2log a c
4

1
161
12 1
3
12

1
4log a b
3

1
log a c
2

2
3

28
3

2

:


a. log a 3b

1
log a log b
2

log 2

a

0; a 2 9b 2 10ab

3b

2 log a 3b

log 2a

b
c

log 2a

a

c
b

0; a 2 9b 2

3b

a 2 6ab 9b 2

2 log 2 log a log b

;

10ab

4ab

a 3b

log a 3b

log 2

log a b.log b c.log c a 1

c
a
b
log 2a ; log 2b ; log 2c
b b
c c
a a
log 2a
log a

b
c

b
c

log 2a

c
a
b
log log 2b log 2c
b
c c
a a
2
a
b

c
b

log a

* log a b.log b c.log c a 1

c
.
b
1

c
b

log a

log a b.log b a

log 3 4

log a

log 3 3 1; log 4

3log6 1,1

3log6 1 1; 7 log6 0,99

7 log6 1 1

c
b

2

1

log 4

1
3

. Ta có :

1
1
log 4 4 1 log 3 4 log 4
3
3
log6 1,1
log6 0,99
3
7
. Ta có :

log 3 4

a/ log 0,4 2 log 0,2 0,34 .

b
c

log a a 1

b
c
a
log a .log b log c
b c
c a
a b

Bài 1

log a2

3 log6 1,1

7 log6 0,99

2

log a2

c
b

2

4ab
1
log a log b
2


2 1

Ta có :

b/ log 5
3

log 0,4 2

0,3 1

3
4

log 3
4

2
.
5

log5

c/ 2log 3 3
5

3
1
4

0

3
1, 0
4

0

1
2

log 5
3

2
1
5

3
4

log 5 1 0
3

2
5

log 3
4

log 3
4

log 3 1 0

2
5

log 5
3

3
4

4

.
2 log

log 5 3 log 5 1

Ta có :

log 0,2 0,3 log 0,4 2

log 0,2 0,3 log 0,2 1 0

5
1
3

Ta có :

log 0,4 1 0

1
log 5
2

log 5 1

3

5

3

log5

2 log

51

1
2

3

log 5 1

20 1
3

0

log 5 3 log 5

1

1
2

d/ log 3 2 log 2 3 .
Ta có :

log 3 1 log 3 2 log 3 3

0 log 3 2 1

log 2 2 log 2 3 log 2 4

1 log 2 3 2

log 2 3 log 3 2

e/ log 2 3 log 3 11 .
Ta có :

1 log 2 3 2
log 3 11 log 3 9

2

log 3 11 log 2 3

2log 2 5 log 1 9

f/ 2

8.

2

Ta có : 2 log 2 5 log 1 9 log 2 25 log 2 9 log 2
2
2

25
9

g/ 4

log2 3 log4

Ta có : 4

5
11

25
92

625
81

648
81

2log 2 5 log 1 9

25
9

2

2

2

log 2

25
9

2log 2 5 log 1 9

8

2

8

2

18 .

log 2 3 log 4

5
11

81.11
5

2

2log 2 3

891
5

1
5
log 2
2
11

90
5

2

log 2 9 log 2

18

4

5
11

2

log 2

log 2 3 log4

5
11

9 11
5

9 11
5
18

81.11
5

25
9


log 3 2 log 1

h/ 9

9

8
9

5.

log 3 2 log 1

Ta có : 9

k/

9

log6 2

1
6

1
log
2

6

3

2 log 3 2 log 9

8
9

3

log 3 2 log 3

8
9

log 3

2.3

3

6
8

8

36
8

3

1
log
2

18 .

6

5

6

log6 2 log6 5

6

log6 10

6

log6

1
10

1
10

3

1
1000

Bài 2. Hãy so sánh :
a/ log 2 10 log 5 30 .
Ta có :

log 2 10 log 2 8 3

log 2 10 log 5 30

log 5 30 log 5 36 3

b/ log 3 5 log 7 4 .
Ta có :

log 3 5 log 3 3 1
log 7 4 log 7 7 1

log 3 5 log 7 4

1
e

c/ 2 ln e3 8 ln .
2 ln e3

Ta có :

1
8 ln
e

2.3 6
8 ln

8 1 9

1
e

2 ln e3

Bài 3
a/ log 1 3 log 3
2

1
2

2.

1

Ta có : log 1 3

log 3

2

log 3

40
8

5

log6 2

1
Ta có :
6

8
9

1
2

0

1
2

log 3

log 3

1
2

1
2

1
1
log 3
2
1
2
1
log 3
2

2

*

log 3

1
2

1
1
log 3
2

2

3

18

5


b/ 4log 7 7log 4 .
5

5

Ta có : 4log 7

log5 7

7log7 4

5

7log5 7.log 7 4

7log5 4

c/ log 3 7 log 7 3 2 .
Ta có : log 3 7 0
d/ 3log

2

5

1
log 3 7

5log5 3

2

1
log 3
2

log 2 5

5log 2 5.log 5 3

5log2 3

log19 log 2 .

1
log 3 log 10 log 3 log 3 10
2
Ta có :
19
361
log19 log 2 log
log
2
4
log 900

f/ log

5

361
4

log

7

5

log 900

1
log 3 log19 log 2
2

log 5 log 7
.
2

2

Ta có :

2

5 log2 3 .

Ta có : 3log 5
e/

log 3 7 log 7 3 log 3 7

7

5. 7

2

log

5

7
2

log 5. 7

log 5 log 7
2

Bài 4. Hãy so sánh :
a. log 3

6
5

log 3

5
6

6
5
Ta có :
5
log 3
6
log 3

b. log 1 9 log 1 17
3

3

5
5
6
log 3
6
log 3

0
0

6
log 3
5

5
log 3
6

6 5
5 6
3 1

log 3

6
5

log 3

5
6


1
1
Ta có :
3
9 17
0

log 1 9 log 1 17
3

3

c. log 1 e log 1
2

2

1
1
2

0

Ta có :

log 1 e

e

log 1

2

2

-GA-RÍT

Bài 1
x2 2x 2 ex

a. y
x2

y

2 x 2 ex

s inx-cosx e 2 x

c. y

y

2 x 2 ex

x2 2 x 2 ex

y'

cosx+sinx e 2 x

ex
ex

ex
ex

e
e

e
e

y

x

y'

x

ex

e

x

ex

e
e

x

ex e

x

x 2

y'

2x
x

2

1

ln x
x

ln x
x

3sin x c osx e 2 x

x

ln x 2 1

e. y

2 s inx-cosx e 2 x

x

d. y ln x 2 1
y

x2 e x

s inx-cosx e 2 x

b. y
y

y'

y'

1 1
.x ln x
x2 x

1 ln x
x2

e

x

ex e

x

4
e

x

e

x 2


f. y

1 ln x ln x

y

1 ln x ln x

ln x 1 ln x
x
x

y'

1 2 ln x
x

Bài 2

:

a. y x 2 ln

y

2

x ln

x2 1

x

2

1

y ' 2 x.ln

x

2

x2 x
2 x2 1

1

2

2 x.ln

x

1

x3
2 x2 1

b. log 2 x 2 x 1
y

log 2 x 2

c. y
y

3

3

x 1

ln 2 x

y'

x 4
x 4

log 2

log

y'

2
3

'

1
3

2
ln x
3

1
x

2
3 x 3 ln x

1
ln 2

16
x 4

2

:

x 4
x 4

16
x 4 ln 2
2

x2 9
x 5

x2 9
x 5

log 3

1

f. y log

y

ln x

x 4
x 4

e. y log 3

y

2

ln 2 x

d. y log 2

y

2x 1
x x 1 ln 2

y'

y'

2
1 2x x 5 x
2
ln 3
x 5

9 x2 9
:
x 5

x 2 10 x 9
x 5 x 2 9 ln 3

x

2 x

1

x

2 x

y'

x 1 1
1
x
:
ln10 16 x x 2 x

x 1
8 x ln10 1

x


Bài 1
a. lim
x 0
lim
x

ln 3 x 1

ln 2 x 1
x

ln 3 x 1

ln 2 x 1
x

0

b. lim
x 0

x

ln 3 x 1
3x
lim
x 0
sin 2 x
2x
2x

ln 3 x 1
0
sin 2 x

c. lim
x 0
lim
x

lim
x

e.

0

e5 x

x

3

0

4x

0

4

2x
3

e3

3

lim e 5
x

0

e5 x 1
2. 5 x

5e 3
2

ex 1
x 1 1

ex 1
x 1 1

a. lim
x 0

ln 4 x 1

e3

2x

lim
x

lim 4

x

0

lim
x

x

0

e5 x

3
2

ln 4 x 1

ln 4 x 1

d. lim
x 0

x

sin 2 x

3x

ln 2 x 1
tan x

ex 1
0
x

lim
x

x 1 1

1.2

ln 2 x 1
0
2
x
2

lim

ln 3 x 1

lim
x

ln 3 x 1
0
3
x
3

lim

2

3 2 1


2x

ln 2 x 1
lim
x 0
tan x

b. lim
x 0
lim
x

e2 x

0

c. lim
x 0

e2 x

lim
x

0

ln 2 x 1
2x
tan x
x
x

2

e3 x
5x

e3 x
5x

e2 x 1
e3 x 1
lim 3
0 5
x 0
5 3x
.2 x
2

lim
x

2 3
5 5

1
5

e3 x 1
x

e3 x 1
e3 x 1
lim 3
3
0
x 0
x
3x

lim
x

d. lim xe
x

1
x

x

1

lim xe

1
x

x

e. lim
x 0
lim
x

0

lim
x

0

lim x e

x

1

ex 1
lim
x
1
x

sin 3 x
x

sin 3 x
x

f. lim
x 0

x

1
x

lim 3
x

0

sin 3 x
3x

3

1 cos5x
x2

1 cos5x
x2

5x
2
lim
2
x 0
4 5x
25 2

Bài 3.
a. lim
x 0

cosx cos3x
sin 2 x

2 sin 2

25
2

1


cosx cos3 x
lim
x 0
sin 2 x

lim
x

2

lim
x

2

sin 2 x

0

x

2

2 sin 2

x

t
2

tan

t
t
2 sin cos
2
2

c. xlim x 2 sin

2

t
.
2

1
cosx

t

Khi x

cos

;t

2

0

2

2

t

2

lim
x

x

1
x

t

;t

1
sin t

t

1
t anx
cosx

lim
t

0

2
t

1 cost
sint

cot t

tan

t
2

t
2

0

3
x 2
x

sin x

4

1
2
3t
t

lim x 2 sin

6t 3

x

3
x

lim 6t 3
t

0

4

x

4

tan

2 2 cos x

d. lim

x

1

t anx=

3
x

3
lim x 2 sin
x
x

lim

4

1
t anx .
cosx
t

x

4 cos x.sin 2 x
lim
x 0
sin 2 x

x

1
t anx
cosx

b. lim
x

2 sin 2 x sin

2 2 cos x
sin x

x

4

x

4

sin x

2 2 cos x
sin x

4

4

;t

0

2 2 cos

2

lim
o

t
2

2

t

2 1 cost+sint
sint

4

t
t
t
2 sin cos
2
2
2
t
t
2 sin cos
2
2

2 tan

4

sin t

2 sin 2

t

4

;x

2 2 cos x

4

2 1 cost+sint
sint

lim

t

t

2

sin
2

t
t
cos
2
2
t
cos
2

2 tan

t
2

2

3



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