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# 对数简介

## 对数是什么?

${2}^{4}=16\phantom{\rule{1em}{0ex}}\phantom{\rule{0.278em}{0ex}}⟺\phantom{\rule{0.278em}{0ex}}\phantom{\rule{1em}{0ex}}{\mathrm{log}}_{2}\left(16\right)=4$

${\mathrm{log}}_{2}\left(8\right)=3$$\phantom{\rule{0.278em}{0ex}}⟺\phantom{\rule{0.278em}{0ex}}$${2}^{3}=8$
${\mathrm{log}}_{3}\left(81\right)=4$$\phantom{\rule{0.278em}{0ex}}⟺\phantom{\rule{0.278em}{0ex}}$${3}^{4}=81$
${\mathrm{log}}_{5}\left(25\right)=2$$\phantom{\rule{0.278em}{0ex}}⟺\phantom{\rule{0.278em}{0ex}}$${5}^{2}=25$

## 对数的定义

${\mathrm{log}}_{b}\left(a\right)=c\phantom{\rule{1em}{0ex}}\phantom{\rule{0.278em}{0ex}}⟺\phantom{\rule{0.278em}{0ex}}\phantom{\rule{1em}{0ex}}{b}^{c}=a$

• $b$$\text{底数}$,
• $c$$\text{指数或者对数}$,
• $a$$\text{幂或者真数}$.

### 看看你的知识掌握地如何

${\mathrm{log}}_{2}\left(64\right)=6$改写为指数形式。

4) 将${\mathrm{log}}_{4}\left(16\right)=2$改写为指数形式。

## 对数的计算

${\mathrm{log}}_{4}\left(64\right)=x$

${4}^{x}=64$
$4$的几次方是$64$呢？我们知道，${4}^{3}=64$，所以${\mathrm{log}}_{4}\left(64\right)=3$

### 看看你的知识掌握地如何

${\mathrm{log}}_{6}\left(36\right)=$

${\mathrm{log}}_{3}\left(27\right)=$

${\mathrm{log}}_{4}\left(4\right)=$

${\mathrm{log}}_{5}\left(1\right)=$

${\mathrm{log}}_{3}\left(\frac{1}{9}\right)=$

## 对变量值的限制

$b>0$在指数表达式中，底数$b$必须是正数。
$a>0$${\mathrm{log}}_{b}\left(a\right)=c$ 意味着 ${b}^{c}=a$. 因为正数的任何次幂都为正，所以 ${b}^{c}>0$, 也就是说 $a>0$.
$b\ne 1$假设 $b$ 可以$1$. 那么对于等式 ${\mathrm{log}}_{1}\left(3\right)=x$，其对应的指数表达式为${1}^{x}=3$。但是$1$的任何次方都是$1$，这个表达式不能成立。因此$b\ne 1$.

## 特殊算法

### 常用对数

${\mathrm{log}}_{10}\left(x\right)=\mathrm{log}\left(x\right)$

### 自然对数

${\mathrm{log}}_{e}\left(x\right)=\mathrm{ln}\left(x\right)$