1
99:59:59,999 --> 99:59:59,999
log以3为底27x的对数

2
99:59:59,999 --> 99:59:59,999
再说一次

3
99:59:59,999 --> 99:59:59,999
只是通过对数运算性质把它展开得到的

4
99:59:59,999 --> 99:59:59,999
这个结果并没有比原式简单

5
00:00:00,000 --> 00:00:04,384
这里要求化简log以3为底27x的对数

6
00:00:04,384 --> 00:00:06,715
这个式子已经够简单了

7
00:00:06,715 --> 00:00:08,523
但是要求用对数运算性质

8
00:00:08,523 --> 00:00:11,110
对它进行转化

9
00:00:11,110 --> 00:00:13,346
结果可能看起来更复杂

10
00:00:13,346 --> 00:00:15,323
下面尽量化一下

11
00:00:15,323 --> 00:00:17,946
可以用对数运算性质转化

12
00:00:17,946 --> 00:00:19,921
因为这里表示

13
00:00:19,921 --> 00:00:22,731
3的几次方可得27x

14
00:00:22,731 --> 00:00:26,444
27x也就是27乘以x

15
00:00:26,444 --> 00:00:30,552
因此用到的对数运算性质是

16
00:00:30,552 --> 00:00:40,110
log以b为底a乘以c的对数

17
00:00:40,110 --> 00:00:41,656
等于

18
00:00:41,656 --> 00:00:48,079
log以b为底a的对数加log以b为底c的对数

19
00:00:48,079 --> 00:00:50,982
这是由指数运算性质推算出来的

20
00:00:50,982 --> 00:00:54,705
也就是 同底数幂相乘时

21
00:00:54,705 --> 00:00:56,421
底数不变指数相加

22
00:00:56,421 --> 00:00:58,121
下面再进一步讲清楚

23
00:00:58,121 --> 00:01:00,910
这部分可能有点难懂 这个例子最重要的是

24
00:01:00,910 --> 00:01:02,656
让大家知道如何去运用对数运算性质

25
00:01:02,656 --> 00:01:04,479
但能了解推导过程当然更好

26
00:01:04,479 --> 00:01:10,571
假设log

27
00:01:10,571 --> 00:01:13,777
log以b为底a乘以c的对数等于x

28
00:01:13,777 --> 00:01:17,587
这部分计算得到x

29
00:01:17,587 --> 00:01:22,079
假设这部分计算得到y

30
00:01:22,079 --> 00:01:26,438
那么log以b为底a的对数等于y

31
00:01:26,438 --> 00:01:32,279
假设这部分计算得到z

32
00:01:32,279 --> 00:01:34,715
那么log以b为底c的对数等于z

33
00:01:34,715 --> 00:01:37,715
现在已知

34
00:01:37,715 --> 00:01:39,715
由这部分 这部分

35
00:01:39,715 --> 00:01:46,829
或者说这部分可得

36
00:01:46,829 --> 00:01:49,823
b的x次方等于a乘以c

37
00:01:49,823 --> 00:01:54,187
由这部分可得

38
00:01:54,187 --> 00:01:56,859
b的y次方等于a

39
00:01:56,859 --> 00:01:59,823
这部分可得

40
00:01:59,823 --> 00:02:02,029
b的z次方等于c

41
00:02:02,029 --> 00:02:04,198
这里同样用绿笔写

42
00:02:04,198 --> 00:02:06,448
写与之等价的表达式

43
00:02:06,448 --> 00:02:08,025
写成指数函数式

44
00:02:08,025 --> 00:02:08,525
或者说指数方程

45
00:02:09,818 --> 00:02:13,685
来替代对数方程

46
00:02:13,685 --> 00:02:16,285
b的z次方等于c

47
00:02:16,285 --> 00:02:18,015
这是意义相同的式子

48
00:02:18,015 --> 00:02:20,352
意义相同的式子

49
00:02:20,352 --> 00:02:23,377
用不同的表达式表示相同的意义

50
00:02:23,377 --> 00:02:25,500
这里用另一种方式表示同种意义

51
00:02:25,500 --> 00:02:28,531
现在已知

52
00:02:28,531 --> 00:02:33,813
a等于它 等于b的y次方

53
00:02:33,813 --> 00:02:36,438
c等于b的z次方

54
00:02:36,438 --> 00:02:41,808
那么

55
00:02:41,808 --> 00:02:43,962
b的x次方等于b的y次方

56
00:02:43,962 --> 00:02:47,275
由前面可知 也就是a

57
00:02:47,275 --> 00:02:49,479
乘以b的z次方

58
00:02:49,479 --> 00:02:52,290
乘以b的z次方

59
00:02:52,290 --> 00:02:54,208
根据指数运算性质可知

60
00:02:54,208 --> 00:02:56,936
根据指数运算性质可知

61
00:02:56,936 --> 00:02:58,744
b的y次方乘以b的z次方

62
00:02:58,744 --> 00:03:04,571
等于

63
00:03:04,571 --> 00:03:06,771
b的y+z次方 换一种颜色写

64
00:03:06,771 --> 00:03:09,905
这是由指数运算性质得到的

65
00:03:09,905 --> 00:03:15,490
b的y+z次方等于

66
00:03:15,490 --> 00:03:19,023
b的x次方 那么一定是x=y+z

67
00:03:19,023 --> 00:03:21,695
x一定是等于y+z

68
00:03:21,695 --> 00:03:24,264
这里比较难懂 不必太担心

69
00:03:24,264 --> 00:03:26,536
首先最重要的是

70
00:03:26,536 --> 00:03:28,459
大家知道怎么用 其次可以思考其意义

71
00:03:28,459 --> 00:03:31,562
也可以代上几个数试试

72
00:03:31,562 --> 00:03:34,675
大家可以看出 对数其实就是指数

73
00:03:34,675 --> 00:03:38,269
你们可能会问 什么意思？

74
00:03:38,269 --> 00:03:41,608
算对数时得到的结果是指数

75
00:03:41,608 --> 00:03:45,141
也就是以b为底数得出a乘以c的那个指数

76
00:03:45,141 --> 00:03:47,223
这里应用这个性质

77
00:03:47,223 --> 00:03:51,669
根据这一性质

78
00:03:51,669 --> 00:04:02,325
log以3为底27乘以x的对数 括起来

79
00:04:02,325 --> 00:04:05,777
等于log以3为底27的对数加log以3为底x的对数

80
00:04:05,777 --> 00:04:10,633
这部分可以算

81
00:04:10,633 --> 00:04:15,152
它表示3的几次方等于27

82
00:04:15,152 --> 00:04:19,125
可以这样写：3的问号次方等于27

83
00:04:19,125 --> 00:04:21,577
3的3次方等于27

84
00:04:21,577 --> 00:04:23,567
3乘3得9 再乘3得27

85
00:04:23,567 --> 00:04:26,275
所以这部分等于3

86
00:04:26,275 --> 00:04:29,171
如果要化简

87
00:04:29,171 --> 00:04:32,392
可能不能称之为化简

88
00:04:32,392 --> 00:04:35,782
要展开或者说用运算性质算

89
00:04:35,782 --> 00:04:40,095
因为原式是一项 现在有两项

90
00:04:40,095 --> 00:04:42,248
原式是这一项

91
00:04:42,248 --> 00:04:45,818
看起来更简单

92
00:04:45,818 --> 00:04:50,198
但是展开后第一项变成3

93
00:04:50,198 --> 00:04:54,613
第一项变成3

94
00:04:54,613 --> 00:04:58,546
再加上log以3为底x的对数

95
00:04:58,546 --> 99:59:59,999
这就是原式的另一种表达方法