# 对数恒等式

## 代数恒等式

### 简化计算

 ${\displaystyle \,\log _{\theta }xy=\log _{\theta }x+\log _{\theta }y}$ 对应到 ${\displaystyle \,\theta ^{x}\theta ^{y}=\theta ^{x+y}}$ ${\displaystyle \log _{\theta }{\frac {x}{y}}=\log _{\theta }x-\log _{\theta }y}$ ${\displaystyle {\frac {\theta ^{x}}{\theta ^{y}}}=\theta ^{x-y}}$ ${\displaystyle \,\log _{\theta }x^{y}=y\log _{\theta }x}$ ${\displaystyle \,({\theta ^{x}})^{y}=\theta ^{xy}}$ ${\displaystyle \log _{\theta }{\sqrt[{y}]{x}}={\frac {\log _{\theta }x}{y}}}$ ${\displaystyle {\sqrt[{y}]{x}}=x^{\frac {1}{y}}}$ ${\displaystyle \,\log _{\theta }-x=\log _{\theta }x+\pi i\log _{\theta }e}$ 欧拉恒等式：${\displaystyle \,e^{\pi i}+1=0}$

### 消去指数

 ${\displaystyle b^{\log _{b}(x)}=x}$ 因为 ${\displaystyle \mathrm {antilog} _{b}(\log _{b}(x))=x\!\,}$ ${\displaystyle \log _{b}(b^{x})=x\!\,}$ 因为 ${\displaystyle \log _{b}(\mathrm {antilog} _{b}(x))=x\!\,}$

### 换底公式

${\displaystyle \log _{\theta }x={\frac {\log _{\phi }x}{\log _{\phi }\theta }}}$

${\displaystyle \log _{a}b={\frac {1}{\log _{b}a}}}$
${\displaystyle \log _{a^{n}}b={{\log _{a}b} \over n}}$
${\displaystyle a^{\log _{b}c}=c^{\log _{b}a}}$

${\displaystyle \log _{a_{1}}b_{1}\,\cdots \,\log _{a_{n}}b_{n}=\log _{a_{\pi (1)}}b_{1}\,\cdots \,\log _{a_{\pi (n)}}b_{n},\,}$

${\displaystyle \pi }$是下标${\displaystyle 1,\ldots ,n}$的任意的排列。例如

${\displaystyle \log _{a}w\cdot \log _{b}x\cdot \log _{c}y\cdot \log _{d}z=\log _{d}w\cdot \log _{a}x\cdot \log _{b}y\cdot \log _{c}z.\,}$

### 和/差公式

${\displaystyle \log _{\theta }(\mathrm {X} \pm \Upsilon )=\log _{\theta }\mathrm {X} +\log _{\theta }\left(1\pm {\frac {\Upsilon }{\mathrm {X} }}\right)}$

### 普通恒等式

 ${\displaystyle \log _{b}(1)=0\!\,}$ 因为 ${\displaystyle b^{0}=1\!\,}$ ${\displaystyle \log _{b}(b)=1\!\,}$ 因为 ${\displaystyle b^{1}=b\!\,}$

## 微积分恒等式

### 极限

${\displaystyle \lim _{x\to 0^{+}}\log _{a}x=-\infty \quad {\mbox{if }}a>1}$
${\displaystyle \lim _{x\to 0^{+}}\log _{a}x=\infty \quad {\mbox{if }}a<1}$
${\displaystyle \lim _{x\to \infty }\log _{a}x=\infty \quad {\mbox{if }}a>1}$
${\displaystyle \lim _{x\to \infty }\log _{a}x=-\infty \quad {\mbox{if }}a<1}$
${\displaystyle \lim _{x\to 0^{+}}x^{b}\log _{a}x=0}$
${\displaystyle \lim _{x\to \infty }{1 \over x^{b}}\log _{a}x=0}$

### 对数函数的导数

${\displaystyle {d \over dx}\ln x={1 \over x}={\ln e \over x}}$

### 积分定义

${\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}dt}$

### 对数函数的积分

${\displaystyle \int \log _{a}x\,dx=x(\log _{a}x-\log _{a}e)+C}$

${\displaystyle x^{\left[n\right]}=x^{n}(\log(x)-H_{n})}$
${\displaystyle x^{\left[0\right]}=\log x}$
${\displaystyle x^{\left[1\right]}=x\log(x)-x}$
${\displaystyle x^{\left[2\right]}=x^{2}\log(x)-{\begin{matrix}{\frac {3}{2}}\end{matrix}}\,x^{2}}$
${\displaystyle x^{\left[3\right]}=x^{3}\log(x)-{\begin{matrix}{\frac {11}{6}}\end{matrix}}\,x^{3}}$

${\displaystyle {\frac {d}{dx}}\,x^{\left[n\right]}=n\,x^{\left[n-1\right]}}$
${\displaystyle \int x^{\left[n\right]}\,dx={\frac {x^{\left[n+1\right]}}{n+1}}+C}$