เอกซ์โพเนนเซียลและลอการิทึม

1. ,n
- (antilogarithm)
-
-

2.
3.
4.
5.
6.

an
an a a a ... a

n
a
n
an

()

1

a,b, a m ,b n a,b 0 m.n

am an am n

am am n
an
(am )n amn

(ab)n anbn

a n an ;b 0
b bn

a1 1 ;a 0
a

a0 1

(32)7 4 17

x, y, z

x 3 y2x 4
yz

()

2

(x 3y 2z 0) 2

4x 2 4x 1 1
2x 2 x 1

xy n y2 = x
y x

Note ! x0 2x 0
x0 2x

(x xx
) x0
2 x 2 x)
1 2
3 x 50
5 2 16
2 = 16
21
12 1
9 39

2 5 2

55

16 -

55 5

()

3

x0 y0 x y xy
x0 y0
xx
yy

)72

10 30

14
2

81
9

xy n
y
nx yn x

xy n y n x xy 0
y nx

nx nx nx nx

()

Note! nn n n
2. n
(Index) n2

2 nx x y

y

4
)

-125
-5
-16
5
) 38

51

4 16

3 1000

3 125

()

5

n n n x n y n xy
xy
xy n y0 nx x
n
Note!
ny y

x n 4 ( 2)4 | 2 | 2
n xn | x | n 5 ( 2)5 2
n xn x

6
) 32 33

3 2 33

3 72
36

7
)22 52 42

43 2
3

()

6

8 (3 5 7 2)( 5 3 2)

x R,x 0,m I,n I ,(m,n) 1 nx R

m1

x n (x n )m (n x )m

2 12 2

( 27)3 ( 27)3 3 27 ( 3)2 9

1 a0

9 (27a 3)6

1 a0 b0

0 (8ab2)3
1
(16a 2b 2 )4

()

7

1 12 27 3

1 1 1 1

2 3(5)3 4(40)3 (1600)6 10 8 3
25

3 53 2 7 3

41

53

5 52 2 3
31 52

()

8

6 2x 3 x
7 x 7 8 2x 1

8 3

)1

22

40
95 57

()

9

76 35
46 5

10 6 2 7
36 27

9 3x 1
) x7

x1 x 2

()

10

3x 4 3x 5 9
4x 1 x 2 x 3

x, y, z
(2x 1y)(x 2y 3z 1) 2

x 4y2z 0 1
2x 3y 1

()

11

x 3y 5 4 x 3y 4 2
x 2y 5

3 46

x2 y3

11

x 2 y2

4 n 2 22n 1 8n 1
23n 2

6n 1 152 n
103 n 4n

()

12

23n 3 22n 1 5 2n 1 6
22n 2 2n 1 6

1

49n 142n n
7n 28n

1

32n 3(24n ) 2n
8n 3(4n )

53n 1 52n 1 361 n 22n 1
5n 1 15 2n 2 25n
n

52n 1

()

13

2 2 2 2...
2 2 2 2 ...
3 3 3 3 ...

()

m1 m2 1 m m m2 1 m1
m2 1 m2 1 m m2 1 m m2 1

100010 , 101000 , 100100 234 , 243 , 423 , 432
x3 1
x14 x3
x

) x2 1

x2

()

15

( 72 3 5) (3 20 50)
3 5( 10 2 5)
(2 3)2
( 5 2)(2 5 1)
(2 3 7)(2 3 7)

()

16

(3 3 1)(3 9 3 3 1)
x 9 11 x
2x 4 x 5 1

()

17

2x 2 7x 1 2x 2 9x 4 1

1 3 x 7 x 16 2x

()

18

9 2x 2x 5
9 2x

x 1 6x
x1

()

19

2x 1 3x 2 4x 3 5x 4
3x 5 8x 1 2x 2 7x 4

()

20

A

3x 2 2 3x 1 3x 10 6 3x 1 14
B

2x 2 6x 11 2 x 2 3x 5 25
AB

()

21

Trick! x 2y

(a b) 2 ab ( a b)
(a b) 2 ab
ab ,a b

2

a b a 2 ab b a b 2 ab

a b 2 ab a b
ab
2

a b a 2 ab b

2

a b a b 2 ab

a b 2 ab ab

20 5 2 6 (3 2) 2 (3)(2) 3 2
21 5 2 6 (3 2) 2 (3)(2) 3 2
22 8 2 15

23 13 2 30

24 x 15 x 22 2 105

()

22

25 3x 1 2 2x 2 3x 2

26 18 8 5

27 5 21 x y xy

()

23

y 3x x { 2, 1, 0,1, 2}

x -2 -1 0 1 2

y 1 1 1 3 9
9 3

y 3x x

(r, 3r ) r

()

(x, 3x ) x (r, 3r )

r y 3x x

y 3x (exponential function)

() 25

28 y 2x

x y y 2x

x -2 -1 0 1 2 3

y1 1 1 2 8

42

y 2x

0 2x x x
y 2x
(0, ) 0

y 2x y
x

()

26

29 y 1 x

2

x y y 1x
2

x -2 -1 0 1 2 3

y 21111

248

y 1x
2

xy y 1x
2

()

27

30 y 2x ,y 3x ,y 10x y 1x
3

y 2x ,y 3x ,y 10x y 1x
3

Note! y ax,a 0 a1 (0,1)

a0 1

a1 y ax

()

28

0a 1 y ax

– xy
– ax ay

()

31 f (x) 2x 1

f f (x) 2x 1 y 2x

()

29

32 g(x) 2x 1

g(x) 2x 1 y 2x

g

(x, y ) | y ax a

a0 a1 1
Note!
y ax
33
1–1
y x2
y ( 1)x y 1x
y (tan 45 )x 2

y 2x
y (cos 30 )x

()

30

0a 1 y 2x
a1 y 2x 1 2
y 1 2x
34
2
y 2x y (tan 60 )x
y 1x
y 1x
2 4
y 22x
y (sin 30 )x

y 5x

y 42x y 3x

()

31

y 4x y 3x
3 4

)

()

32

) y 2x 3 y 2x 3

y4 1x y 10x 3
2

()

33

25x 1 am an mn
5
( )
52x 5 1

2x 1

x 1
2

am am ab m0

3x 1 5x 1 ( )
x10
x1

()

35 x 6 1 x 216
x
2x 4

2x 4

36

()

37 x 22x 5x 0.0025

38 3 3 3x 16 x
8 81

39 x 23x 1 6x 55x 1 15x

40 x 4(9)x 1 3 22x 1

()

35

41 4x y 128 32x y 81 y

42 xy 64x 2 2x 4
|x y |
2x y 256 3

43 5x 1 10x 2 x

32 32

()

36

x

44 11 2 30 2 6 5 x

45 3 5 2x 4 x2 2x

4 35

46 9x 81y 1z 1x x0 1192y
28 100z 2x

()

37

47 x 5x 5x 1 5x 2 155 5

48 x 8x 8x 1 8x 2 228

49 x 3x 1 3x 3x 1 45

()

38

50 x 17x 4 16(17x 3) 175x
51 x 72x 4 7x 5 0
52 x 4x 1 64 2x 5

()

39

53 2x x

2 3 42 3 1 0

54 5 5x 5 x 26

55 3x 32 x 4 3

()

56 6 25x 11 23x 3 2x 25x 1
57 3x2 2x 3x2 1 9x 1 27 0
58 3 4x 6x 2 9x 0

()

59 12x 2(3x ) 9(4x ) 18 0

60 2 2 22x2 x2 2x 2 5 4x x 2 2x

61 x 0 8x 8 4x 2x 3 x

()

62 x 256x (2x 6)4 8x

x 1 x1

9(2 2) 22
63 A x / 3 27[22x1 32 ] A

()

64 A 3(1 2x ) 9(2 x ) 244 A

65 A 5 1 x2 4x 1 5 4x x2 126

A 5 2 x2 4x 1

()

66 1 1 7
2x 22x 2x 2x 22x 2x 2

67 A 4x 2x 6 3 2x 4 3 4x 2 3

A

()

a 1( * -
)
am an
0 a 1( ) mn

am an
mn

32x 5 35x 6 1 2x 5 1 5x 6
3 3

2x 5 5x 6 2x 5 5x 6
1
x 3x 1 3x
1 x1
3
3

53x 2 23x 2

53x 2 23x 2 53x 2 23x 2
23x 2 23x 2 53x 2 53x 2
5 3x 2 2 3x 2
2 1 1 5

5 3x 2 50 20 2 3x 2
2 2 5 5
0
3x 2 0 x 3x 2
x 2 2
3 3

()

68 (0.4)2x 8 2

5

69 (0.04)5x2 x 8 625
70 5(2x 2) (0.8)(5x 2)

71 3 3x2 2x 4 2x 15

43

()

72 1 x2 2x 8 1 x 12

24

73 2 x(1 x) 9

34

74 4(2x2 4x 5) 1

32

75 1 x2 3x 5 1 x 10

28

()

76 2x2(x 3) 8(2 x )
3

77 53x 2 73x 2
78 35x 9x2 27
79 (sin 30 )3x2 4x 1 (sin 30 )x2 x 1

()