1
00:00:00,000 --> 00:00:02,200
1/2乘以log以2为底32的对数

2
00:00:00,000 --> 00:00:01,200
1/2仍然在外面

3
00:00:00,000 --> 00:00:01,000
10/4-3/4=7/4

4
00:00:00,000 --> 00:00:01,333
3乘以1/4 减去3/4

5
00:00:00,000 --> 00:00:01,000
log 32

6
00:00:00,000 --> 00:00:02,666
log以2为底32的对数减去 对吗？

7
00:00:00,000 --> 00:00:01,000
乘以log 32

8
00:00:00,000 --> 00:00:01,600
也是这节课开始的

9
00:00:00,000 --> 00:00:01,600
但请明白其中要点

10
00:00:00,000 --> 00:00:01,000
再见

11
00:00:00,000 --> 00:00:01,733
减去 这里8的1/2次方

12
00:00:00,000 --> 00:00:01,000
减去1/4

13
00:00:00,000 --> 00:00:02,133
减去1/4log以2为底8的对数

14
00:00:00,000 --> 00:00:02,333
减去log以2为底根号8的对数

15
00:00:00,000 --> 00:00:01,800
变成整个式子的系数

16
00:00:00,000 --> 00:00:01,800
只是移了指数的位置

17
00:00:00,000 --> 00:00:01,400
因为1/2乘进来了

18
00:00:00,000 --> 00:00:01,400
因为这里是相除

19
00:00:00,000 --> 00:00:01,000
它等于1/2

20
00:00:00,000 --> 00:00:01,200
它等于括号log

21
00:00:00,000 --> 00:00:01,000
对吗？

22
00:00:00,000 --> 00:00:01,200
所以它又等于

23
00:00:00,000 --> 00:00:01,600
把外面的1/2乘进去

24
00:00:00,000 --> 00:00:01,200
有可能算错了

25
00:00:00,000 --> 00:00:01,000
然后可以

26
00:00:00,000 --> 00:00:01,866
现在这里是相除 对吗

27
00:00:00,000 --> 00:00:03,400
由这集视频一开始讲的那个性质得出的

28
00:00:00,000 --> 00:00:02,200
等于1/2log以2为底32的对数

29
00:00:00,000 --> 00:00:02,533
等于1/2乘以log以2为底8的对数

30
00:00:00,000 --> 00:00:01,000
算算看

31
00:00:00,000 --> 00:00:02,000
这时可以用另一个性质

32
00:00:00,000 --> 00:00:01,600
这是5/2 减去 这是3

33
00:00:00,000 --> 00:00:01,400
这里忘了写底数

34
00:00:00,000 --> 00:00:01,600
这里还有一个根号

35
00:00:00,000 --> 00:00:01,000
那个性质

36
00:00:00,000 --> 00:00:01,000
除以根号8

37
00:00:00,000 --> 00:00:01,333
除以根号8 对吗

38
00:00:00,000 --> 00:00:01,000
欢迎回来！

39
00:00:00,000 --> 00:00:02,600
这次讲最后两个对数运算性质

40
00:00:00,000 --> 00:00:01,000
这个

41
00:00:00,000 --> 00:00:01,400
我认为这个性质

42
00:00:00,000 --> 00:00:02,800
从某种程度上说是最显而易见的

43
00:00:00,000 --> 00:00:03,000
但就算你觉得不那么明显也别气馁

44
00:00:00,000 --> 00:00:01,600
可能需要反思一下

45
00:00:00,000 --> 00:00:02,400
我鼓励大家把这些运算性质

46
00:00:00,000 --> 00:00:01,200
亲自演练演练

47
00:00:00,000 --> 00:00:03,000
因为只有这样才能真正把它弄明白

48
00:00:00,000 --> 00:00:02,600
学数学不单单是为了考试及格

49
00:00:00,000 --> 00:00:01,266
或者说考试得A

50
00:00:00,000 --> 00:00:02,200
学数学是为了真正理解它

51
00:00:00,000 --> 00:00:02,600
也就能在以后的生活中运用它

52
00:00:00,000 --> 00:00:02,800
而不必在用到的时候再重学一遍

53
00:00:00,000 --> 00:00:02,400
接下来讲的对数运算性质是

54
00:00:00,000 --> 00:00:02,000
A乘以log以B为底C的对数

55
00:00:00,000 --> 00:00:01,266
A乘以这个式子

56
00:00:00,000 --> 00:00:02,600
等于log以B为底C的A次方的对数

57
00:00:00,000 --> 00:00:01,000
真妙

58
00:00:00,000 --> 00:00:01,400
看看成立不成立

59
00:00:00,000 --> 00:00:01,000
假设

60
00:00:00,000 --> 00:00:02,000
3乘以log以2为底8的对数

61
00:00:00,000 --> 00:00:01,200
根据这个性质

62
00:00:00,000 --> 00:00:01,000
这里等于

63
00:00:00,000 --> 00:00:02,200
log以2为底8的3次方的对数

64
00:00:00,000 --> 00:00:01,000
也就相等-

65
00:00:00,000 --> 00:00:01,866
等于 我们可以算一下

66
00:00:00,000 --> 00:00:01,400
算算这边是多少

67
00:00:00,000 --> 00:00:02,866
3乘以log log以2为底8的对数是几？

68
00:00:00,000 --> 00:00:02,000
刚才这边我犹豫了一下

69
00:00:00,000 --> 00:00:02,200
是因为每次要算一个式子

70
00:00:00,000 --> 00:00:03,400
我总是潜意识的会用对数指数法则去算

71
00:00:00,000 --> 00:00:01,000
这里应避免

72
00:00:00,000 --> 00:00:01,266
好 回来继续算

73
00:00:00,000 --> 00:00:01,400
这部分是多少？

74
00:00:00,000 --> 00:00:01,533
2的几次方等于8？

75
00:00:00,000 --> 00:00:01,866
2的3次方等于8 对吗？

76
00:00:00,000 --> 00:00:01,000
所以是3

77
00:00:00,000 --> 00:00:02,066
这里还有个3 所以3乘以3

78
00:00:00,000 --> 00:00:01,866
那么这边应该也等于9

79
00:00:00,000 --> 00:00:01,066
如果它等于9

80
00:00:00,000 --> 00:00:01,200
可知这个性质

81
00:00:00,000 --> 00:00:02,200
至少对这个例子是成立的

82
00:00:00,000 --> 00:00:03,200
如果不清楚是否在任何情况下都成立

83
00:00:00,000 --> 00:00:01,400
而想看推导证明

84
00:00:00,000 --> 00:00:01,800
可以收看另一集视频

85
00:00:00,000 --> 00:00:01,800
那一集讲的更理论性

86
00:00:00,000 --> 00:00:01,600
而现在最重要的是

87
00:00:00,000 --> 00:00:01,200
明白怎么应用

88
00:00:00,000 --> 00:00:02,000
来看 2的9次方是多少？

89
00:00:00,000 --> 00:00:01,200
算出来会很大

90
00:00:00,000 --> 00:00:01,533
我们知道 应该是-

91
00:00:00,000 --> 00:00:02,400
因为上一集视频里我们算过

92
00:00:00,000 --> 00:00:01,333
2的8次方等于256

93
00:00:00,000 --> 00:00:01,933
那么2的9次方应该是512

94
00:00:00,000 --> 00:00:01,133
2的9次方是512

95
00:00:00,000 --> 00:00:01,733
如果8的3次方也是512

96
00:00:00,000 --> 00:00:01,266
就对了 是吗？

97
00:00:00,000 --> 00:00:02,066
因为log以2为底512的对数

98
00:00:00,000 --> 00:00:01,000
等于9

99
00:00:00,000 --> 00:00:01,533
8的3次方是几呢？

100
00:00:00,000 --> 00:00:01,000
64 对

101
00:00:00,000 --> 00:00:02,066
8的平方是64 那么立方是

102
00:00:00,000 --> 00:00:01,666
算算 4乘8 2 进3 6乘8

103
00:00:00,000 --> 00:00:01,000
等于512

104
00:00:00,000 --> 00:00:01,000
对的

105
00:00:00,000 --> 00:00:02,000
也可以用其他方法验证

106
00:00:00,000 --> 00:00:01,533
可以验证8的3次方

107
00:00:00,000 --> 00:00:01,533
是否等于2的9次方

108
00:00:00,000 --> 00:00:01,000
等不等呢？

109
00:00:00,000 --> 00:00:01,000
8的3次方

110
00:00:00,000 --> 00:00:02,466
等于2的3次方的3次方 对吗？

111
00:00:00,000 --> 00:00:01,266
只是把8替换了

112
00:00:00,000 --> 00:00:01,800
由指数运算法则可知

113
00:00:00,000 --> 00:00:01,400
2的3次方的3次方

114
00:00:00,000 --> 00:00:01,133
等于2的9次方

115
00:00:00,000 --> 00:00:02,000
实际上是指数运算性质

116
00:00:00,000 --> 00:00:02,200
一个底数的乘方后再乘方

117
00:00:00,000 --> 00:00:01,800
可以把两个指数相乘

118
00:00:00,000 --> 00:00:01,600
积作为最终的指数

119
00:00:00,000 --> 00:00:02,400
本质上也就是指数可以相乘

120
00:00:00,000 --> 00:00:01,600
这个指数运算性质

121
00:00:00,000 --> 00:00:02,200
导出了这个对数运算性质

122
00:00:00,000 --> 00:00:01,400
这里不再详细讲

123
00:00:00,000 --> 00:00:01,000
推导过程

124
00:00:00,000 --> 00:00:02,000
另外有一集视频专门讲

125
00:00:00,000 --> 00:00:01,000
推导证明

126
00:00:00,000 --> 00:00:02,400
下面要讲的对数运算性质是

127
00:00:00,000 --> 00:00:04,000
过后会举几个例子复习一下前面学的几个性质

128
00:00:00,000 --> 00:00:01,600
如果你常用计算器

129
00:00:00,000 --> 00:00:02,800
这个性质可能是最有用的一个了

130
00:00:00,000 --> 00:00:01,000
为什么呢

131
00:00:00,000 --> 00:00:02,133
假设是log以B为底A的对数

132
00:00:00,000 --> 00:00:03,866
等于log以C为底A的对数除以log以C为底B的对数

133
00:00:00,000 --> 00:00:01,800
为什么说常用计算器

134
00:00:00,000 --> 00:00:02,000
这个性质就很有用呢？

135
00:00:00,000 --> 00:00:02,266
假设你去上课 有个小测验

136
00:00:00,000 --> 00:00:01,600
老师允许用计算器

137
00:00:00,000 --> 00:00:01,400
要用计算器算出

138
00:00:00,000 --> 00:00:01,733
log以17为底357的对数

139
00:00:00,000 --> 00:00:01,600
你会赶紧找计算器

140
00:00:00,000 --> 00:00:02,800
有没有log以17为底这个键 找不到

141
00:00:00,000 --> 00:00:02,200
因为计算器上压根就没有

142
00:00:00,000 --> 00:00:01,000
这个键

143
00:00:00,000 --> 00:00:01,400
可能会看到log键

144
00:00:00,000 --> 00:00:01,000
或者ln键

145
00:00:00,000 --> 00:00:02,266
你要知道 计算器上的log键

146
00:00:00,000 --> 00:00:01,000
底数是10

147
00:00:00,000 --> 00:00:01,200
ln键的底数是e

148
00:00:00,000 --> 00:00:01,666
如果不知道e是什么

149
00:00:00,000 --> 00:00:02,133
也没关系 它等于2.71几几

150
00:00:00,000 --> 00:00:01,000
一个数字

151
00:00:00,000 --> 00:00:01,200
很奇妙的数字

152
00:00:00,000 --> 00:00:01,800
以后的课程会讲到它

153
00:00:00,000 --> 00:00:02,466
现在 计算器上只有两种底数

154
00:00:00,000 --> 00:00:02,200
要想算出其他底数的对数

155
00:00:00,000 --> 00:00:01,600
就用到这个性质了

156
00:00:00,000 --> 00:00:02,200
所以如果考试碰到这个题

157
00:00:00,000 --> 00:00:01,600
你就可以很自信了

158
00:00:00,000 --> 00:00:01,000
它等于

159
00:00:00,000 --> 00:00:01,200
把笔换成黄色

160
00:00:00,000 --> 00:00:01,600
以表示自信的感觉

161
00:00:00,000 --> 00:00:01,866
log 底数选e或10都可以

162
00:00:00,000 --> 00:00:01,000
就等于

163
00:00:00,000 --> 00:00:01,733
log以10为底357的对数

164
00:00:00,000 --> 00:00:02,066
除以log以10为底17的对数

165
00:00:00,000 --> 00:00:02,000
这时就可以在计算器上

166
00:00:00,000 --> 00:00:01,000
输入357

167
00:00:00,000 --> 00:00:01,000
按下log键

168
00:00:00,000 --> 00:00:01,000
得到某个数

169
00:00:00,000 --> 00:00:01,000
然后清除

170
00:00:00,000 --> 00:00:02,200
或者会用计算器上的括号

171
00:00:00,000 --> 00:00:01,000
也可以做

172
00:00:00,000 --> 00:00:01,733
不会的话就再输入17

173
00:00:00,000 --> 00:00:01,666
按log键 得到某个数

174
00:00:00,000 --> 00:00:01,666
然后相除 得出答案

175
00:00:00,000 --> 00:00:04,266
所以如果喜欢用计算器 这就是个非常有用的性质

176
00:00:00,000 --> 00:00:02,666
再说一遍 这里不会讲的太深入

177
00:00:00,000 --> 00:00:02,800
它对我来说是最有用的一个性质

178
00:00:00,000 --> 00:00:01,400
但是它不完全是

179
00:00:00,000 --> 00:00:03,000
它显然是由指数运算性质推出来的

180
00:00:00,000 --> 00:00:01,800
但一两句话也说不清

181
00:00:00,000 --> 00:00:02,200
如果你不相信它是成立的

182
00:00:00,000 --> 00:00:01,400
可以看推导证明

183
00:00:00,000 --> 00:00:01,400
但是先不管这些

184
00:00:00,000 --> 00:00:02,400
这个性质你在以后的生活里

185
00:00:00,000 --> 00:00:01,000
会经常用到

186
00:00:00,000 --> 00:00:01,600
我工作中仍然在用

187
00:00:00,000 --> 00:00:02,000
你要知道对数用处很多

188
00:00:00,000 --> 00:00:01,400
下面做几个例子

189
00:00:00,000 --> 00:00:02,600
把复杂的式子转化成简单形式

190
00:00:00,000 --> 00:00:01,000
假设是log

191
00:00:00,000 --> 00:00:01,066
以2为底根号

192
00:00:00,000 --> 00:00:01,000
多少呢

193
00:00:00,000 --> 00:00:01,400
32除以3次根号下

194
00:00:00,000 --> 00:00:01,000
算了 根号

195
00:00:00,000 --> 00:00:01,000
除以根号8

196
00:00:00,000 --> 00:00:02,200
怎么转化成简单形式呢？

197
00:00:00,000 --> 00:00:01,000
想一想

198
00:00:00,000 --> 00:00:01,000
它等于

199
00:00:00,000 --> 00:00:02,466
到底是横着写 还是竖着写呢

200
00:00:00,000 --> 00:00:01,000
竖着写吧

201
00:00:00,000 --> 00:00:01,600
它等于log以2为底32

202
00:00:00,000 --> 00:00:02,866
除以根号8 括起来 的1/2次方 对吗

203
00:00:00,000 --> 00:00:01,800
由对数运算性质可知

204
00:00:00,000 --> 00:00:01,466
刚学的第3个性质