对数,真数,底数
【转】https://www.cnblogs.com/chenxi188/p/11050016.html
对数及运算法则
1.对数源于指数,是指数函数反函数
因为:y = ax
所以:x = logay
2. 对数的定义
【定义】如果 N=ax(a>0,a≠1),即a的x次方等于N(a>0,且a≠1),那么数x叫做以a为底N的对数(logarithm),记作:
其中,a叫做对数的底数,N叫做真数,x叫做 “以a为底N的对数”。
2.1对数的表示及性质:
1.以a为底N的对数记作:logaN
2.以10为底的常用对数:lgN = log10N
3.以无理数e(e=2.71828...)为底的自然对数记作:lnN = logeN
4.零没有对数.
5.在实数范围内,负数无对数。 [3]在虚数范围内,负数是有对数的。
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注: 自然对数的底数 e :https://www.guokr.com/article/50264/
细胞分裂现象是不间断、连续的,每分每秒产生的新细胞,都会立即和母体一样继续分裂,一个单位时间(24小时)最多可以得到多少个细胞呢?答案是:
当增长率为100%保持不变时,在单位时间内细胞种群最多只能扩大2.71828倍。 数学家把这个数就称为e,它的含义是单位时间内,持续的翻倍增长所能达到的极限值。
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3.对数函数
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D162/sign=c26ae4e5cfea15ce45eee40f84013a25/b8389b504fc2d56244645a8de11190ef76c66c66.jpg)
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D49/sign=a933c77dd22a283447a637025ab5f7f1/b03533fa828ba61e61b91f4f4734970a304e597f.jpg)
![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D33/sign=5f60285976cf3bc7ec00cbefd0006afd/b999a9014c086e0625fceb1f04087bf40ad1cb0a.jpg)
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D58/sign=472bfe85ab6eddc422e7b4f338db7366/503d269759ee3d6d1c2457ab45166d224e4adef4.jpg)
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D49/sign=a933c77dd22a283447a637025ab5f7f1/b03533fa828ba61e61b91f4f4734970a304e597f.jpg)
![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D33/sign=a7931d8cd700baa1be2c41b8461066ac/1c950a7b02087bf4327f8789f4d3572c11dfcf3b.jpg)
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D49/sign=a933c77dd22a283447a637025ab5f7f1/b03533fa828ba61e61b91f4f4734970a304e597f.jpg)
4.对数运算法则(rule of logarithmic operations)
对数运算法则,是一种特殊的运算方法。指 积、商、幂、方根 的对数的运算法则
由指数和对数的互相转化关系可得出:
![](https://gss1.bdstatic.com/-vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D178/sign=6629e608dc62853596e0d626a8ee76f2/18d8bc3eb13533fa7ff20fd4a1d3fd1f40345b80.jpg)
![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D176/sign=318278d107f41bd5de53ecf367db81a0/eac4b74543a982268360b4dd8382b9014b90eba3.jpg)
![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D120/sign=a43ff0aa3da85edffe8cfa21795509d8/c2cec3fdfc039245ce8678f58e94a4c27d1e251c.jpg)
![](https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D130/sign=7573df47a0773912c0268162c8188675/54fbb2fb43166d22cc3a8c6e4f2309f79152d2ff.jpg)
![](https://img2018.cnblogs.com/blog/1525006/201906/1525006-20190620092930483-1130486746.png)
5.对数公式
5.1基本知识
① ;
![](https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D76/sign=19dfd7afaad3fd1f3209a03c314e17ba/f9198618367adab4d1b95fc289d4b31c8601e4c2.jpg)
![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D39/sign=a57883a126a446237acaa36b9922ef7e/2cf5e0fe9925bc31943af84358df8db1ca137082.jpg)
![](https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D40/sign=ae857ff0b651f819f525024adbb4bf32/b17eca8065380cd706b45b36a744ad3458828147.jpg)
![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D115/sign=4afd137ba2af2eddd0f14de8b8110102/ca1349540923dd546abaca15df09b3de9d824885.jpg)
5.2恒等式及证明
![](https://img2018.cnblogs.com/blog/1525006/201906/1525006-20190620094124291-2080696369.png)
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