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Tính đơn điệu của hàm số 1... Đưa về cùng cơ số và logarit hóa:[r]

CHUYÊN ĐỀ 7: PHƯƠNG TRÌNH, BẤT PHƯƠNG TRÌNH MŨ – LOGARIT

(TRẦN THÀNH SANG)  Cơng thức cần nhớ

an = an

1

; a0 = ;

m m n n

a  a ( m; n nguyên

dương , n > 1)

 Các quy tắc:

ax.ay = ax+y (a.b)x =ax.bx x

a x y

a y a

 

x x

a a

x b b        x y  y x x.y

a  a a

 = logaN  a = N

logax = b  x= ab

 Đặc biệt : alogax = x ; loga ax = x ;

loga1 =

 Các qui tắc biến đổi : với a , B , C > ; a 

ta có:

loga(B.C) = logaB + logaC

loga

B C

   

  = logaB  logaC

loga B =



logaB

 Công thức đổi số : với a , b , c > ; a , c 

1 ta có :

logca.logab = logcb 

log bc log ba

log ac



0 < a, b  : logab =

1 log ab

Chú ý : log10x = lg x ; logex = ln x  Hàm số mũ - logarit

 Hàm số mũ : y = ax với a > ; a 

TXĐ : D = R MGT : (0; + )

+ a > ; h/s đồng biến : x1 > x2  x

a

> ax2

+ < a < ; h/s nghịch biến : x1 > x2  x

a

< ax2

 Hàm số Logarit: y = logax với a > 0; a 

TXĐ : D = (0 ; + ) MGT : R

+ a > ; h/s đồng biến : x1 > x2 >  logax1 >

logax2

+ < a < 1;h/s ngh biến: x1 > x2 >  logax1

<logax2

 Đạo hàm hàm số mũ - logarit

(ex) / = ex

( ax) / = ax.lna

(lnx) / =

1

x x (0;+)

(logax) / =

1 x ln a

( au)/ = u/.au.lna

( eu)/ = u/.eu

(lnu)/ =

u u



(logau )/ =

u u ln a

1 Phương trình mũ - logarit :

a Đưa số logarit hóa:

f (x)

a = ag(x)  f(x) = g(x) v(x)

u =  ( u(x) 1 ).v(x) = f (x)

a = b ( với b > )  f(x) = logab

logaf(x) = logag(x) 

f (x) g(x) f (x) g(x)

 



  

log f (x)a b a



 

 

  f(x) = ab

logu(x)v(x)

= b   

v(x) ; u(x) ; u(x) b

v(x) u(x)

  



    

BÀI TẬP THAM KHẢO

1. 2x4 3

2.

2 6

2

2x x 16

3. 32x3 9x23x5

4. 2x2 x 41 3 x

5. 52x + 1 – 52x -1 = 110

6.

5 17

7

32 128

4

x x

x x

 

  

7. 2x+ 2x -1 + 2x – 2 = 3x – 3x – 1 + 3x - 8. (1,25)1 – x = (0,64)2(1 x)

9. 22x + 5 + 22x + 3 = 12 10. 334 92 2



x x

11. 5x 8x −x1

=500

12. 2x(√x2+4− x −2)=4(√x2+4− x −2)

13.2x+1 + 2x+2 = 5x+1 + 3.5x 14.2x - = 3

15.3x + 1 = 5x – 16.3x – 3 = 5x27x12 17. 2x2 5x25x6

18.52x + 1- 7x + 1 = 52x + 7x 19. log3x+log3(x+2)=1

20. log2(x2−3)−log2(6x −10)+1=0 21. ln(x+1)+ln(x+3)=ln(x+7)

22. logx4+log(4x)=2+logx3

23. log4[(x+4)(x+3)]+log4

x −2

x+3=2

24. log√3(x −2)log5x=2 log3(x −2) 25.log4(x + 2) – log4(x -2) = log46 26.lg(x + 1) – lg( – x) = lg(2x + 3)

27.log4x + log2x + 2log16x = 28.log4(x +3) – log4(x2 – 1) = 29.log3x = log9(4x + 5) + ½ 30.log4x.log3x = log2x + log3x –

31.log2(9x – 2+7) – = log2( 3x – + 1) 32.log(x2 – x -2 ) < 2log(3-x)

33.2 – x + 3log52 = log5(3x – 52 - x) 34.log3(3x – 8) = – x

35. log7x=log3(√x+2)

b Đặt ẩn phụ:

.ab f (x) +.ab f (x) +  = ; đặt : t = af (x) t >

.af (x)+.bf (x)+  = a.b = 1; đặt: t = af (x);

1 t=bf (x)

.a2f (x)+. 

f (x) a.b +

.b2f (x) = ; đặt: t =

f (x) a b

     

BÀI TẬP

1. 34x8 4.32x527 0 2. 4x 5.2x 4

3. 4x 2.2x1 3

4. 6.9x13.6x6.4x 0 5. 1 2

  

x x

6. 16x17.4x16 0

7. 22 2 9.2 2 0

  

x x

8. 32x1 9.3x 6

9. 6.9x13.6x6.4x 0 10.

1

5

2

2 5

x x

   

  

   

   

11. x 53 x 20

12. 4 15 4 15

x x

   

13.  6  6 10

x x

   

14. 4x2−3x+2

+4x2+6x+5=42x2+3x+7+1

15. 23x−6 2x−

23(x −1)+

12

2x=1

16. 9x+2 (x −2)3x+2x −5=0

17.

2x

100x=6 (0,7)

x

+7

18. (1

3)

2

x+3

(13)

1

x+1 = 12

19. 4x1 2x1 2x2 12

  

20. 22x21 9.2x2x22x2 0

21. -7 3x-1

+√1-6 3x+9x+1=0

22. 4x-13 6x

+6 9x=0

23. 12 3x+3 15x-5x+1=20

24. 32x-1

=2+3x-1

25. (√6-√35)x+(√6+√35)x=12

26. 4x-6 2x+1

+32=0

27. 9x−

(263 ).3

x

+17=0

28. 22x+1−2x+3−64

=0

29. (√2−√3)x+(√2+√3)x=4

30. (7+4√3)x−3(2−√3)x+2=0

31. 4x2+1

+6x

2

+1

=9x

2

+1 32. 2x2−5x+6

+21− x2=2 26−5x+1

33. log16x+logx4=3

34. logx2−log4x+7

6=0

35. log3x −2 log2x=−2+logx

36. 4

+log2x+

2

2−log2x=1

37. log4x8−log2x2+5

2=0

38. 3√log3x −log3(3x)−1=0 39.

x −1¿3=7

x −1¿2+log2¿

log22

¿

40. log22(4x)+log2x

8 =8

41. log3

(3x)+

logx3=7

42.

x −1¿2=log2(4x)

1

2log√2(x+3)+log4¿ 43.

log√2+1√x2−3x+2+log√2−1√x −1=log3−2√2(4x+8)

44. log2(3x−1)log2(2⋅3x−2)=2 45. log2(2x)logx(2x)=log41

2

46. log2

x

2+log2(4x)=3 47. log5x5

x+log5

c Tính đơn điệu hàm số 1. 25x+10x=22x+1

2. 4x−2 6x

=3 9x

3. 4 3x

−9 2x=5

x

2

4. 125x

+50x=23x+1

5.  

2 2

1

2x -2x x  x 1

6. 3x + 5x = 6x + 2 7. 1+82x

=3x

8. 32x −3+(3x −10)3x −2+3− x=0

9. −2x2

− x

+2x−1=(x −1)2

10.3x + 4 x = 5x 11.3x – 12x = 4x 12. log2x+√2x+2=2

13.

2x+√1+log2x

=1

14. log2(x2−4)+x=log2[8(x+2)] 15. log22x+ (x-5)log2x-2x+6=0 16. log2(x+3log6x)

=log6x

17. 2log2(x+1)

=x

18. log4

√5(x

−2x −2)=log2(x2−2x −3)

19. x2

+3log2x

=xlog25

20. log3

2

x+(x −4)log3x − x+3=0 2 Bất phương trình mũ - logarit

 af (x)> ag(x) 

f (x) g(x) a f (x) g(x) a

 

  

  

 af (x) > b

b  có nghiệm x

b > 0, a>1  f(x) > logab

b>0, < a <  f(x) < logab  af (x) < b

b  pt vơ nghiệm

b > 0, a >  f(x) < logab

b > 0, 0<a< f(x) > logab

logaf(x) > logag(x) (0 < a  1)

f(x) > g(x) >

(a1)[ f(x)  g(x) ] >

logaf(x) > b

a >  f(x) > ab < a <  < f(x) < ab

logaf(x) < b

a >  < f(x) < ab < a <  f(x) > ab

 

v(x) u(x) >

 u(x) >

 u(x)v(x)<  u(x) > [ u(x) 1 ].v(x) < a Đưa số logarit hóa:

1. 33xx−2x+2

−2x ≤1

2. (√5+2)x-1≥(√5−2)

x-1

x+1

3. (√10+3)

x−3

x−1

<(√10−3)

x+1

x+3

4. 2√x12

−2x≤2

x −1

5. 9x+9x+1+9x+2<4x+4x+1+4x+2

6. 2|21x+1|≥

1 23x+1

7. (x2

−2x+1)

x−1

x+1≥1

8. (x2−1)x2+2x

>|x2−1|3

9. 3x+1

+5x+3≤3x+4+5x+2

10. 16x – 4 ≥ 8 11.

2

1

9

x

      

12.

6

9x 3x

13. 6

4x x 

14.

2

4 15

3

1

2

2

x x

x

 



 

  

 

15. 52x + > 5x 16.

2 logsin

3

 



x x

17. log2(x2−16)≥log2(4x −11) 18. |log3x −2|<1

19. √log321x −− x3<1

20. log2

3

log3|x −3|≥0

21. log1

2[

log2(3x+1)]>−1

22. logx(5x2−8x+3)>2

23. logx3x −1

24. (0,08)logx −0,5x

≥(5√2

2 )

logx−0,5(2x −1)

25. 0,12logx−1x≥

(5√33)

logx −1(2x −1)

26. |1+logx2004|<2

27. loga(35− x

3)

loga(5− x) >3

28. (4x−12 2x

+32)log2(2x −1)≤0 29. logx2(

4x −2

|x −2|)≥

1

30. log1

3

√2x2−3x+1

>

log1

3

(x+1)

31. log24x −log1 2

(x83)+9 log2( 32

x2)<4 log1 2 x

32. log1

5

(x2−6x+8)+2 log5(x −4)>0

33. log1

2[

log4(x2−5)

]>0

34. log2x(x

2

−5x+6)<1

35. 5log3x −x2

<1

36. log3(3x−1)

x −1 ≥1

37. 12log1

2

(x −1)>log1

2

(1+√3 x −2)

38.

¿ xlogyz

+zlogyz

=512

ylogzx

+xlogzy

=8

zlogzx

+ylogxz

=2√2

¿{ {

¿

b Đặt ẩn phụ: 1. (13)2x+3

(13)

2

x+1>12

2. 9x-2 3x

<3

3. (3+√5)2x-x

2

+(3−√5)2x-x

2

-21+2x-x2

0

4. 22√x+3− x −6

+15 2√x+3−5<2x

5. 251+2x− x2

+91+2x − x

2

34 52x − x2

6. 3log3

x−18 xlog31x

+3>0

7. 32x−8 3x+√x+4−9 9√x+4

>0

8. (13)x−1−(1

9)

x

>4

9. √9x−3x+2

>3x−9

10. 9√x2

−3+1

+3<28 3√x2−3−1

11. 4x2+1

32x−4 3x+1≤0

12. 4x2

+x+1−2x+2

+1≤0

13. log2

2 + log2x ≤ 14. log1/3x > logx3 – 5/2 15. log2 x + log2x ≤ 16.

1

1 log xlogx 

17. 16

1 log 2.log

log

x x

x





18.

4

3

log (3 1).log ( )

16

x

x 

 

1. y y x x         

2. 2

3 ( )( 8)

8

y

x y x xy

x y            3. y y x x          4.

3 11

3 11

x y x y y x            5.

2 36 36

y x y x       

6. 2

2

3

y

x y x

x xy y

           7. 4 32 x x y y        8.

4

4 144

y x y x         9.

2 20

5 50

y x y x        10.

2 17

3.2 2.3

y x y x          11.

3

3

13. 2

6

log log

x y x y        14.

 2 

2

3

log

log log

x y x y           15.

log log

6 y

x y x

x y         

16. 2

6

log

log log

x y x y         

17.    

2

3

3

log log

x y

x y x y

           18. 2 log

2 log

x y x y        19. log

log

9 y y x x         20. 2 2

log log 16

log log

y x x y x y          21.    

log 2

log 2

x y x y y x            22. 2 log log

3 10

log log

y x x y x y          23. 32 logy