对数函数换底公式推倒

比眉伴天荒 2023-06-13 04:46 50阅读 0赞

## 简介： ##

对于

![format_png][]  且![format_png 1][]  ，有![format_png 2][]

### 推导过程 ###

**法一：**若有[对数][Link 1]

![format_png 3][]   ，设 ![format_png 4][]  ，

![format_png 5][]  。

则根据对数的基本公式

![format_png 6][]

和

![format_png 7][]

及

![format_png 4][]  ,

![format_png 8][] 可得 ![format_png 9][]

则有 ![format_png 10][]

证毕。

**法二：**若有对数

![format_png 11][] ，则 ![format_png 12][]  ，且 ![format_png 13][]

于是

![format_png 14][]

两边取以c为底的对数得

![format_png 15][] ，

![format_png 16][] ，即![format_png 17][] 。

证毕。

**法三：**若有对数

![format_png 11][]  ，则![format_png 12][]  ，且![format_png 18][] ，于是![format_png 19][]

即![format_png 20][]    从而![format_png 21][]

证毕。

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[Link 1]: https://baike.baidu.com/item/%E5%AF%B9%E6%95%B0
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