เอกซ์โพเนนเซียลและลอการิทึม

2x2 3x 1 3 7 x2 8x 3

80 3 7 2

81 22x 2x 1 8 0

82 34x 3 26(32x 3) 1

()

50

83 23x2 2 5(22x2 ) 23x2 2x2 1
84 8x 18x 2 27x 0

()

51

85 A 3 5x2 23x 3 5 x5

A 53

86 A (x 2)x2 2 (x 2)2x 10 x2

A

()

52

87 A x | 22x 2x 2 2x 1 32
2

A

a a0 a1 x,y | y ax

1–1

x,y | x ay

x ay y f (x) y loga x

“” ”

x ay y loga x

loga x x0
x,y | y loga x

()

53

y loga x a y ax

yx

y loga x

()

(logarithm function)

x,y | y loga x a a0 a1

Note! 1

y loga x

1–1

logarithm m,n 0

a,b

loga 1 0 loga m loga n log2 1 0 log3 5 log3 2
loga a 1 log2 2 1
loga (mn) log3(10) log3(5 2)

loga m loga m loga n log2 3 log2 3 log2 6
n 6

logby ax x logb a log9 27 log32 33 3 log3 3 3
y 2 2

a loga m m 2log2 3 3
a logb m
logb a m logb a 2 5log3 5 log3 2

logb a logm a log3 2 log2 3 log 2 log 3 1
logm b log 3 log 2

1 log2 3 1
loga b log3 2

()

55

Note! (common logarithms) log e
log 2 log10 2
(natural logarithms) log

e
ln x loge x

88
) log5 125

log2 1
128

log3 3 3

log4 2 log4 32

log2 80 log2 5

()

56

89 log9 7 log3 7 1.771

90 log10 3.2 log10 2 0.3010

91

log1000 3

4 log3 81

log9 1 0

log3 1 5
243

logb a c

p logq r

()

57

92 loga x 1 , logb x 1 , logc x 1 logd x 1
2 3 4 5
logx abcd

93 a,b,c 1 loga d 30, logb d 50 logabc d 15

logc d

94 7log7 52 5 log2 4 3 2 log9 33

()

58

14

95 81 27 3log5 3 log7 9
log9 36

96 A log4 8 log9 3 log6 4 log6 9 B 161 log4 3 81AB

97 log3 5 1.465 log27 15 log3 5 1.465

()

59

98 log10 28 log 1 325 log 1 91

10 100

99 log2 1 log3 2 log4 3 ... log8 7

100 log(log 81) log(log 27)

log(3)log3 4 log(4)log4 3

101 8eln6 ln2

(lne2)(eln 3)(eln 4 )

()

60

102 x 9 3x x log3 2 1

103 a 3 b4 c6 d12 logd (abc)

104 log6 5 a log36 125 log5 36

()

61

105 x y 2x5y 1
5x 1 2y 2

106 2x 2x 1 2x 2 4x 4x 1 4x 2

()

62

(antilogarithm)

log10 3 10

log 3

log10 N log N

N N 0 10n 1 N 0 10 n
1200 1.2 103
0.035 3.5 10 2

N

107 log 5760 log 0.576 log 5.76 0.7604

log 5760 log 5.76 103

log 5.76 log103

0.7604 3 0.7604 3
log 5760 3.7604
log 0.576 3.7604

log 5.76 10 1

log 5.76 log10 1

0.7604 1 0.7604 1
log 0.576 0.2396

0.2396

log N N log N

NN (antilogarithm) logN

A B B log A

()

63

108 log 5.71 0.7566 log N 3.7566 N

log 3.71 0.5694 log 8.32 0.9201

) log 37100

log 0.00371

log 832

log 0.0832

()

log 2.56 0.4082 N

) log N 0.4082

log N 3.4082

log N 0.5918

log 0.0000000196 log1.96 0.2923

log 1200 log1.2 0.0792

log 0.00652 log 6.52 0.8142

()

65

Antilog( 3.5918) log 2.56 0.4082

log N 1.1463 N log 7.14 0.8537

log 63.7 1.9041 10 1.0959

()

66

logarithm

loga x1 loga x2 x1 x2 a0 a 1

log a,b loga x logb x

x1 0 a,b 1

- log
- log

109 log x 2 log 4 log 32 x

110 log(3x 2) log(x 1) 1

()

67

111 log(x 2) log(x 1) 1

112 log2(x 2) log2(x 1) 2
113 log3(2x 3) log3(x 2) 2

114 log(4x 2 4) log(x 2 1) 2

logx 10

()

68

115 log(2x x 4) x(1 log 5)

116 log4(2x2 x 28) 3x log8 2

117 log (4 x) log2(9 4x ) 1
2

()

69

118 x log4 x log9 3 log3 9

119 x log2 x 3 log4 x 6 log8 x 9 8log4 9

120 x 2 log3 0.5 log0.5 x log3 4

()

70

121 x y y 1 logy 2x a 2y b
2 log2 b a
x
2a log2 b
1 log2 b a
2

a log2 b
2

122 log4 2 log3(1 log2 a) 1 a 22x a a2 3a 4
2
x

123 x (log2 x)2 8 log16 x 3 0

()

71

124 (log x 3)2 log(0.1x 10) 0

125 x log2 x logx 2 5
2

126 log3 x 1 logx 9

()

72

127 x 2 log3 x 2 log 2 9 3 0
x

128 logy x 4 logx y 4 logy x 3

129 2 2 2log(x 2) log(x 3) log 2

()

73

130 x log 7 log 3 log 7
7 x 2 xlog 3 log 7 log 2

131 x 2 x 2logx log2 log x 12

132 x x log x 10

()

133 x x logx 100x

134 x log(x 10) 2 log(x 10) 1 2

log(x 1) log(x 1)

135 logy x logx y 2 x 2 y 20 log2(2x 2y )

()

75

logarithm ) loga m loga n

log mn

a 1( loga m loga n

0 a 1( ) mn

- log
- log

136 log16 x log4 x log2 x 7

137 log2(2x 4) log2(x 1)

()

76

138 log1(x 3) log1(2x 6)

22

139 1 log2 (x2 2) 1 log2 (4x 1)
3 3

140 (0.4)log0.5(2x2 2x 4) (0.4)log0.5( 3x 3)

()

77

141 log1(log3(x 1)) 1

2

142 log0.2 x log5(x 2) log0.2 3

143 log2(x 4) 0

()

78

144 log(3x 4) log(x 1) 1

145 log1(x 2 x) log1(24 x)

22

146 log1(2x 1) log1 (2x 1)

24

()

79

147 log1 log4(x 1) 1

2

148 (2logx2 ) (5logx ) 0.0025

149 (a,b) 3(2log x ) 2 x log 4 ab

()

80

n(t) n0(1 r)t

n(t) t
n0
r 55%

150 8 ,100

)

) 12

()

81

151 7

B(n) B0(1 r)n

B(n) n
B0
r 5

152
) 30,000

()

82

) 10 100,000

) 20,000
30,000

Note! 6 3
1
B(n) n
B0 B(n) B0(1 r )mn
r m
m
6 m2
() 3 m4

m 365

83

(Inflation) 10 20
20 100 10 100

5 10 t 100,000
V (t) V0e rt
V (t)
V0 t
r
10
153

n(t) n0ert

n(t) t
n0
r

()

154 10
500

(half - life) h

m(t) m0e rt

m(t) t

m0 - 210 300

r ln 2
h

155 - 210

- 210 1

()

85

- 210 200
- 210 50

(sound level)

10 12 1

10 log I 10 12
I0

I
I0

()

86

156 100

– pH

pH pH = -log[H3O+ ]
[H3O+ ]
–

pH 7
pH 7
pH 7

157 3.16 10 8

pH –

()

87