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# 对数属性简介

(当对数均已被定义时，$M$, $N>0$，且$0，无论$M$, $N$, 以及$b$为任何值，这些法则均适用.)

## 乘积法则: ${\mathrm{log}}_{b}\left(MN\right)={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)$‍

### 示例：用乘积法则展开对数

$\begin{array}{rl}{\mathrm{log}}_{6}\left(5y\right)& ={\mathrm{log}}_{6}\left(5\cdot y\right)\\ \\ & ={\mathrm{log}}_{6}\left(5\right)+{\mathrm{log}}_{6}\left(y\right)& & \text{乘法法则}\end{array}$

### 示例：用乘积法则缩写对数

$\begin{array}{rlrl}{\mathrm{log}}_{3}\left(10\right)+{\mathrm{log}}_{3}\left(x\right)& ={\mathrm{log}}_{3}\left(10\cdot x\right)& & \text{乘法法则}\\ \\ & ={\mathrm{log}}_{3}\left(10x\right)\end{array}$

### 看看你的知识掌握地如何

1)展开${\mathrm{log}}_{2}\left(3a\right)$

2)缩写${\mathrm{log}}_{5}\left(2y\right)+{\mathrm{log}}_{5}\left(8\right)$

## 除法法则：${\mathrm{log}}_{b}\left(\frac{M}{N}\right)={\mathrm{log}}_{b}\left(M\right)-{\mathrm{log}}_{b}\left(N\right)$‍

### 示例：用商法则展开对数

$\begin{array}{rlr}{\mathrm{log}}_{7}\left(\frac{a}{2}\right)& ={\mathrm{log}}_{7}\left(a\right)-{\mathrm{log}}_{7}\left(2\right)& \text{除法法则}\end{array}$

### 示例：用商法则缩写对数

$\begin{array}{rlrl}{\mathrm{log}}_{4}\left({x}^{3}\right)-{\mathrm{log}}_{4}\left(y\right)& ={\mathrm{log}}_{4}\left(\frac{{x}^{3}}{y}\right)& & \text{除法法则}\end{array}$

### 看看你的知识掌握地如何

3)展开${\mathrm{log}}_{b}\left(\frac{4}{c}\right)$.

4)缩写$\mathrm{log}\left(3z\right)-\mathrm{log}\left(8\right)$.

## 幂法则: ${\mathrm{log}}_{b}\left({M}^{p}\right)=p{\mathrm{log}}_{b}\left(M\right)$‍

### 示例：用幂法则展开对数

$\begin{array}{rlrl}{\mathrm{log}}_{2}\left({x}^{3}\right)& =3\cdot {\mathrm{log}}_{2}\left(x\right)& & \text{幂规律}\\ \\ & =3{\mathrm{log}}_{2}\left(x\right)\end{array}$

### 示例：用幂法则缩写对数

$\begin{array}{rlrl}4{\mathrm{log}}_{5}\left(2\right)& ={\mathrm{log}}_{5}\left({2}^{4}\right)& & \text{幂规律}\\ \\ & ={\mathrm{log}}_{5}\left(16\right)\end{array}$

### 看看你的知识掌握地如何

5) 展开${\mathrm{log}}_{7}\left({x}^{5}\right)$.

6) 缩写$6\mathrm{ln}\left(y\right)$.

## 挑战题

7) 下列哪个表达式与${\mathrm{log}}_{b}\left(\frac{2{x}^{3}}{5}\right)$等值？

8) 下列哪个表达式与$3{\mathrm{log}}_{2}\left(x\right)-2{\mathrm{log}}_{2}\left(5\right)$等值？