If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# 证明对数属性

${\mathrm{log}}_{b}\left({b}^{c}\right)=c$

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

$\begin{array}{rlrl}{\mathrm{log}}_{b}\left(MN\right)& ={\mathrm{log}}_{b}\left({b}^{x}\cdot {b}^{y}\right)& & \text{代入}\\ \\ & ={\mathrm{log}}_{b}\left({b}^{x+y}\right)& & \text{指数性质}\\ \\ & =x+y& & {\mathrm{log}}_{b}\left({b}^{c}\right)=c\\ \\ & ={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)& & \text{代入}\end{array}$

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

$\begin{array}{rlrl}{\mathrm{log}}_{b}\left(\frac{M}{N}\right)& ={\mathrm{log}}_{b}\left(\frac{{b}^{x}}{{b}^{y}}\right)& & \text{代入}\\ \\ & ={\mathrm{log}}_{b}\left({b}^{x-y}\right)& & \text{指数性质}\\ \\ & =x-y& & {\mathrm{log}}_{b}\left({b}^{c}\right)=c\\ \\ & ={\mathrm{log}}_{b}\left(M\right)-{\mathrm{log}}_{b}\left(N\right)& & \text{代入}\end{array}$

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

$\begin{array}{rlrl}{\mathrm{log}}_{b}\left({M}^{p}\right)& ={\mathrm{log}}_{b}\left({\left({b}^{x}\right)}^{p}\right)& & \text{代入}\\ \\ & ={\mathrm{log}}_{b}\left({b}^{xp}\right)& & \text{指数性质}\\ \\ & =xp& & {\mathrm{log}}_{b}\left({b}^{c}\right)=c\\ \\ & ={\mathrm{log}}_{b}\left(M\right)\cdot p& & \text{代入}\\ \\ & =p\cdot {\mathrm{log}}_{b}\left(M\right)& & \text{乘法交换律}\end{array}$

$\begin{array}{rlrl}{\mathrm{log}}_{b}\left({M}^{p}\right)& ={\mathrm{log}}_{b}\left(M\cdot M\cdot \text{…}\cdot M\right)& & \text{指数的定义}\\ \\ & ={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(M\right)+\text{…}+{\mathrm{log}}_{b}\left(M\right)& & \text{乘法法则}\\ \\ & =p\cdot {\mathrm{log}}_{b}\left(M\right)& & \text{叠加即乘法}\end{array}$