luythua mu logarit

7/26/2019 LuyThua Mu Logarit

1/6

Chuyn 5

Hm S Ly Tha. Hm S M & HmS Lgarit

1. Ly Tha

A. Kin Thc Cn Nh

1. Cc nh ngha.Ly tha vi s m nguyn dng: an = a.a...a

n tha s

(a R, n N).

Ly tha vi s m 0: a0 = 1 (a = 0).Ly tha vi s m nguyn m: an = 1

an (a = 0, n N).

Cn bc n: bl cn bc n ca a bn =a.Lu : Khi n l th ac ng mt cn bc nl n

a.

Khi n chn th a 0 c hai cn bc nl

n

a.Ly tha vi s m hu t: amn = nam (a >0; m, n Z; n 2).Ly tha vi s m thc: a = lim

n+arn

a >0; (rn) Q; lim

n+rn=

.

2. Cc tnh cht ca ly tha vi s m thc.Cho hai s a,b >0 v, l nhng s thc tu . Ta c

a.a =a+. a

a =a.

(a) =a. (ab) =a.b.

ab

=

a

b.

Nu a >1 th a > a > . Nu 0 < a a < .

Nu >0 th 0 < a < b

a < b.

Nu b.

B. Bi Tp

5.1. Tnh gi tr cc lu tha sau

a)(0, 04)1,5 (0, 125) 23 . b)

1

16

0,75+

1

8

43

.

c)2723 +

1

16

0,75 250,5. d)(0, 5)4 6250,25

2

1

4

1 12

.

e)810,75 +

1

125

13

1

32

35

. f) 102+

7

22+7.51+

7

.

g) 423 431 .223. h) 625 + 46 31 + 26 31 26.5.2. Rt gn cc biu thc sau

a) x

54 y+ xy

54

4

x + 4

y . b)

a13

b + b

13

a

6

a + 6

b.

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Nguyn Minh Hiu

c)

a b

4

a 4b

a 4ab4

a + 4

b. d)

a b3

a 3b a + b3

a + 3

b.

e)

a23 1

a23 + a

3 + a3

3

a43 a3 .

f)

a + b3

a + 3

b 3

ab

:

3

a 3

b2

.

g) a 1a34 + a

12

.

a + 4

a

a + 1 .a

14 + 1. h) a +

b32

a12

a12 b 12

a12

+ b

12

a12

b12

23

.

5.3. Hy so snh cc cp s saua) 3

10v 5

20. b) 4

13v 5

23.c)3600 v5400. d) 3

7 +

15v

10 + 3

28.

5.4. Tnh A =

a + b + c + 2

ab + bc +

a + b + c 2ab + bc, (a, b, c >0, a + c > b)

2. Lgarit

A. Kin Thc Cn Nh

1. nh ngha. = logab

a =b (a,b >0; a

= 1).

2. Tnh cht. loga1 = 0. logaa= 1. alogab =b. loga(a) =.Khi a >1 th logab >logac b > c. Khi 0< a logac b < c.

3. Quy tc tnh. loga(bc) = logab + logac. loga bc = logab logac. loga 1b = logab. logab =logab. loga n

b= 1

nlogab. logab= logac.logcb.

logab= 1logba . logab= 1 logab.4. Lgarit thp phn v lgarit t nhin.

Lgarit thp phn: L lgarit c c s a= 10. K hiu: log xhoc lg x.Lgarit t nhin: L lgarit c c s a= e. K hiu: ln x.

B. Bi Tp

5.5. Tnha)log3

4

3. b)2log27log 1000. c)log258.log85.d) log 45 2log3. e)3log2log416 + log 1

22. f)log248 13 log227.

g)5 ln e1 + 4ln

e2

e

. h) log 72 2log 27256

+ log

108. i)log 0, 375 2log0, 5625.5.6. n gin biu thc

a) log24 + log2

10

log220 + log28 . b)

log224 12 log272log318 13 log372

. c)

log72 +

1

log57

log7.

d) logaa2. 3

a. 5

a4

4

a . e)log5log5

5

5

... 5

5 n du cn

. f)92log34+4log812.

g)161+log45 + 412log23+3log55. h)

81

14 1

2log94 + 25log1258

49log72. i)72

49

12log79log76 + 5log54

.

5.7. So snh cc cp s sau:a)log3

65 v log3

56 . b)log 12 evlog 12 . c)log210vlog530.

d) log53vlog0,32. e)log35 vlog74. f)log310vlog857.

5.8. Tnh log41250theo a, bit a = log25.

5.9. Tnh log54168theo a, b, bit a= log712, b= log1224.

5.10. Tnh log14063theo a , b, c, bit a = log23, b= log35, c= log72.

5.11. Tnh log 325135theo a, b, bit a= log475, b= log845.

5.12. Chng minh rng ab + 5 (a b) = 1, bit a= log1218, b= log2454.5.13. Cho y = 10

11log x , z= 10

11log y . Chng minh rng x = 10

11log z .

5.14. Cho a,b,c >0. Chng minh rng (abc)a+b+c

3 aabbcc.

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Chuyn 5. Hm S Ly Tha. Hm S M & Hm S Lgarit

3. Hm S Ly Tha. Hm S M & Hm S Lgarit

A. Kin Thc Cn Nh

1. Hm s lu tha.Dng: y = x ( R).

Tp xc nh:

Nunguyn dng th D = R.Nu= 0hoc nguyn m th D= R\ {0}.Nukhng nguyn th D = (0; +).

o hm: y= x1.Tnh cht: (Xt trn (0;+))

>0: Hm s lun ng bin. 0 1: Hm s lun ng bin.a 1 0< a 1: Hm s lun ng bin.a 1 0< a 0: ax =b x= logab.

2. Bt phng trnh m c bn.Dng: ax > b (0< a = 1).Cch gii:

b 0: S= R.b >0, a >1: ax > b x >logab.

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Nguyn Minh Hiu

B. Phng Phng Gii C Bn

a v cng c s. t n ph.Ly lgarit hai v. S dng tnh n iu ca hm s m.

C. Bi Tp

5.18. Gii cc phng trnh saua)22x1 = 3. b)2x

2x = 4.c)2x

2x+8 = 413x. d)3x.2x+1 = 72.e)32x1 + 32x = 108. f)2x + 2x+1 + 2x+2 = 3x + 3x1 + 3x2.

g)

3 + 2

2x+1

=

3 222x+8. h)5 26x23x+2 5 + 26 1x22 = 0.5.19. Gii cc bt phng trnh sau

a)2x2+3x 5x+1 5x+2. d)2x + 2x+1 + 2x+2 0, 25.128

x+17x3 . h)2x

2

.7x2+1 0. b)32.4x + 1 < 18.2x.c)5x + 51x >6. d)

2 +

3x

+

2 3x >4.5.22. Gii cc phng trnh sau

a)

5 26

x

+

5 + 2

6

x

= 10. b) (B-07)

2 1

x

+

2 + 1

x 22 = 0.

c)7 + 3

5x + 5.7 3

5x = 6.2x. d) 5 + 2

6x

+ 5 2

6x

= 10.e)

7 + 4

3x 32 3x + 2 = 0. f)26 + 153x + 27 + 43x 223x = 1.

5.23. Gii cc phng trnh saua)3.4x 2.6x = 9x. b)2.16x+1 + 3.81x+1 = 5.36x+1.c)4x+

x22 5.2x1+

x22 6 = 0. d)5.2x = 7

10x 2.5x.

e)27x + 12x = 2.8x. f) (A-06)3.8x + 4.12x 18x 2.27x = 0.5.24. Gii cc bt phng trnh sau

a)27x + 12x 4.3x + 3.2x. b)4x

2+x + 21x2 2(x+1)2 + 1.

c)52x+1 + 6x+1 >30 + 5x.30x. d)52x103x2 4.5x5


5/6

Chuyn 5. Hm S Ly Tha. Hm S M & Hm S Lgarit

5.29. Gii cc phng trnh saua)22x 2x + 6 = 6. b)32x + 3x + 7 = 7.c)27x + 2 = 3 3

3x+1 2. d)7x1 = 6log7(6x 5) + 1.

5.30. Gii cc phng trnh saua)2x

2

= 3x. b)2x24 = 3x2.

c)5x.8x1x = 500. d)8

xx+2 = 4.34x.

5.31. Gii cc phng trnh saua)3x

2

= cos 2x. b)2|x| = sin x.c)2x1 + 2x

2 x.2x1 2x2x = (x 1)2. d)22x+1 + 232x = 8log3(4x24x+4) .

5. Phng Trnh & Bt Phng Trnh Lgarit

A. Kin Thc Cn Nh

1. Phng trnh lgarit c bn.Dng: logax= b (0< a = 1).

Cch gii: log

a

x= b

x= ab.

2. Bt phng trnh lgarit c bn.Dng: logax > b (0< a = 1).

Cch gii: a >1: logax > b

x > ab.

0< a b 0< x < ab.Lu . Cc dng logax b;logax < b;logax bda vo du c cch gii tng ng.

B. Phng Phng Gii C Bn

a v cng c s. t n ph.S dng tnh n iu ca hm s lgarit.

C. Bi Tp

5.32. Gii cc phng trnh saua)log3(x

2) = 2. b)log3(5x + 3) = log3(7x + 5).

c)log2 x2 1= log 12 (x 1). d)log2x + log2(x 2) = 3.e)log2

x2 + 8

= log2x + log26. f) log3(x + 2) + log3(x 2) = log35.

g)log3x + log4x= log5x. h)log2x + log3x + log4x= log20x.

5.33. Gii cc bt phng trnh saua)log8(4 2x) 2. b)log3

x2 + 2

+ log 1

3(x + 2) < 0.

c)log 15

(3x 5)> log 15

(x + 1). d)log2(x + 3) < log4(2x + 9).

5.34. Gii cc phng trnh saua)log2

x2 + 3x + 2

+ log2

x2 + 7x + 12

= log224. b)log

x3 + 8

= log (x + 58) + 12log

x2 + 4x + 4

.

c) 12

log2(x + 3) + 14

log4(x 1)8 = log24x. d) 32 log 14 (x + 2)2 3 = log 1

4(4 x)3 + log 1

4(x + 6)

3.

e)log2

x + 1 log 12

(3 x) log8(x 1)3 = 0. f) log 12 (x 1) + log 12 (x + 1) log 12 (7 x) = 1.g)log2 8 x2 + log 12 1 + x +1 x 2 = 0. h)log2(4x + 15.2x + 27) + 2log2 14.2x3 = 0.

5.35. Gii cc phng trnh saua)log2

x x2 1 + 3log2 x + x2 1= 2. b) (A-08)log2x1 2x2 + x 1+logx+1(2x 1)2 = 4.

c)log2

x x2 1 .log3 x +x2 1= log6 x x2 1.5.36. Gii cc bt phng trnh sau

a) (A-07)2log3(4x 3) + log 13

(2x + 3) 2. b)log 12

x + 2log 14

(x 1) + log26 0.c) (D-08)log 1

2

x23x+2x 0. d)log0,5 x+12x1 >1.

e) log2

3.2x1 1

x 1. f)

log2(1 3log27x) 1log2x

log3log 1

5

x2 + 1 x.

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Nguyn Minh Hiu

5.38. Gii cc phng trnh saua)log22 x 3log2x + 2 = 0. b)log 12 x + log

22 x= 2.

c)2log2x log3x= 2 log x. d)log2x3 20logx + 1 = 0.e)

log3x +

4 log3x= 2. f) log2(2x + 1) .log2

2x+1 + 2

= 2.g)log3(3

x + 1) .log3

3x+2 + 9

= 3. h)log2(5x 1) .log4(2.5x 2) = 1.

5.39. Gii cc bt phng trnh sau

a)log22(2x + 1) 3log(2x + 1) + 2 > 0. b)log29(x 1) 3log3(x 1) + 1 0.c)logx14 1 + log2(x 1). d)log2(2x 1)log 12

2x+1 2> 2.

e)log4(19 2x)log2 192x

8 1. f) log5(4x + 144) 4log52< 1 + log5

2x2 + 1

.

5.40. Gii cc bt phng trnh sau

a)

log2x +

logx2 43 . b) 3

log 12

x + log4x2 2> 0.

c)

log22 x + log 12

x2 3> 5 log4x2 2. d) log22 x + log2x4 8> log2 x24.5.41. Gii cc bt phng trnh sau

a)log2x64 + logx216 3. b)logx(125x) .log25x > 32 + log25 x.c) (C-2012)log2(2x). log3(3x)> 1. d)log 1

3x + 1 4log

212

x 11 x. b)1 +15x 4x.c)1 + 2x+1 + 3x+1 2.3log x2+2.e)log7x 0, h phng trnh

ex ey = ln (1 + x) ln (1 + y)y x= a c nghim duy nht.

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luythua mu logarit

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