1
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你好

2
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让我们一起做一下关于对数的练习

3
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好，我们来快速回顾一下什么是对数

4
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如果我写，以x为底的a对数是

5
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等于，随便说吧，n

6
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这是什么意思

7
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这就是说x的n次方等于a

8
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我想我们已经知道这点

9
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我们已经在对数视频上知道这点

10
00:00:40,150 --> 00:00:42,860
所以要意识到这在你计算的时候很重要

11
00:00:42,860 --> 00:00:49,170
一个对数式，比方说以x为底的a的对数，它的答案是

12
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当你计算时，你得到的是，一个幂

13
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这个n真的就是个幂

14
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这也等同于这个东西

15
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你也可以写成这种形式

16
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因为n等于这个东西，你可以

17
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有点混乱，写成x的以x为底的a的对数次方

18
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等于a

19
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我所做的就是用这个等式代替n

20
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我写成这种形式是因为我想你

21
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真的明白这个写法

22
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当你计算时，这个对数，它

23
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就是一个指数

24
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还有，我们将要把这个写法

25
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这就是所有对数

26
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性质的来历

27
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我想要下一步做的是

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我想在对数性质上

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再玩味玩味

30
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然后我就此总结，然后

31
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擦干净

32
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不过我想展示以下人们当初

33
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怎样发现这个东西的

34
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拿x当例子，让我换一种颜色

35
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我认为这会有趣些

36
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举例说，x的l次方等于a

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如果我们写成对数的形式

38
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我们可以写成

39
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以x为底的a的对数等于l，对吧

40
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我只是抄写一次上面的信息

41
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现在，让我换一种颜色

42
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如果我说x的m次方等于b

43
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一样的道理，我只要改变字母

44
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这就是说以x为底的b的对数是

45
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等于m，对吧

46
00:02:43,730 --> 00:02:46,280
我只是做跟这一行同样的事

47
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我要换一种颜色

48
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让我们继续，看有什么发生

49
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让我换一种颜色

50
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打个比方，x的n次方

51
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但是你会看到

52
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x的n次方等于a乘以b

53
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x的n次方等于a乘以b

54
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这就是说

55
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以x为底的ab的对数

56
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那我们可以怎样做呢？

57
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我们从这里开始

58
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x的n次方等于a乘以b

59
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那我怎样改写呢？

60
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a是这个

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b是这个，对吧？

62
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那我们改成这样子

63
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我们已知x的n次方等于a

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a是这个

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x

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x的l次方

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x的l次方

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乘以b

69
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b是x的‘m次方，对吧？

70
00:04:04,740 --> 00:04:07,380
现在不做任何花俏的事

71
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x的l次方乘以x的m次方等于什么呢？

72
00:04:09,320 --> 00:04:13,730
从指数运算，我们知道

73
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当你的底数一样

74
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指数不同时，你只要将指数相加

75
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那就是等于，让我换一种中立的颜色

76
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我不知道这样说是否正确

77
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你会得到

78
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当你有相同的底数，又在做乘法

79
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你只要将指数相加

80
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等于x的l+m次方

81
00:04:32,390 --> 00:04:33,870
我想换一种颜色，因为这更好

82
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l,l+m

83
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经常换颜色有点烦

84
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你知道我的意思的

85
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所以，x的n次方等于x的l+m次方

86
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让我把x写在这儿

87
00:04:49,790 --> 00:04:51,350
oh，我想用绿色

88
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x的l+n次方

89
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那我们知道什么呢？

90
00:04:54,050 --> 00:04:58,980
我们知道x的n次方等于x的l+m次方

91
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对吧

92
00:05:00,220 --> 00:05:02,510
我们有相同的底数

93
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这些指数一定相同

94
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所以我们得知n=l+m

95
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这有什么用呢?

96
00:05:21,270 --> 00:05:23,590
我只是围着指数玩玩

97
00:05:23,590 --> 00:05:25,840
我有什么结论了么?

98
00:05:25,840 --> 00:05:27,590
我想你就要看到了

99
00:05:27,590 --> 00:05:31,140
那,n的另一种写法是什么呢?

100
00:05:31,140 --> 00:05:34,510
我们说,x的n次方等于a乘以b

101
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我事实上跳了一步

102
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返回去,x的n次方

103
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等于a乘以b

104
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这意味着以x为底的a乘以b的对数是n

105
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你知道的

106
00:05:45,170 --> 00:05:45,890
我不知道

107
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我希望你没意识到我不是在倒带

108
00:05:47,880 --> 00:05:52,360
我只是一开始忘了将这个写下来

109
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不过没关系

110
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那,n等于什么

111
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n的另一种写法是什么?

112
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n的另一种写法是这样的

113
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以x为底的a乘以b的对数

114
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那我们现在知道如果我们用n代替这个

115
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得到以x为底的a乘以b的对数

116
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这个等于什么呢?

117
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这等于l

118
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l的另外一种表示方法如下

119
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它等于以x为底a的对数,加上m

120
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那什么是m呢

121
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m在这儿

122
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也就是以x为底,b的对数

123
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这样我们就得到我么第一个对数的性质

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以x为底的a乘以b的对数

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就等于以x为底的a的对数乘以以x为底的b的对数

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希望这证明给你看

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还有,如果你想知道这为什么直觉上

128
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会想到对数不过是指数

129
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我会留下这个视频

130
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在下一个视频,我会证明另一个

131
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对数的性质

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以后见