# 對數定律

## 定律

${\displaystyle a}$${\displaystyle M}$${\displaystyle N}$都係一個正數，而且${\displaystyle a\neq 1}$，重有${\displaystyle x}$係未知數。有以下實數解：

1. ${\displaystyle \log _{a}1=0}$
2. ${\displaystyle \log _{a}a=1}$
3. ${\displaystyle \log _{a}a^{x}=x}$
4. ${\displaystyle a^{\log _{a}x}=x,x>0}$
5. ${\displaystyle \log _{a}MN=\log _{a}M+\log _{a}N}$
6. ${\displaystyle \log _{a}{\bigl (}{\frac {M}{N}}{\bigr )}=\log _{a}M-\log _{a}N}$
7. ${\displaystyle \log _{a}M^{N}=N\log _{a}M}$

(1)

(2)

(3)

(4)

(5)

(6)

(7)

## 一般基數同自然基數分別

${\displaystyle \log 1=0}$ ${\displaystyle \ln 1=0}$
${\displaystyle \log 10=1}$ ${\displaystyle \ln e=1}$
${\displaystyle \log 10^{x}=x}$ ${\displaystyle \ln e^{x}=x}$
${\displaystyle 10^{\log x}=x,x>0}$ ${\displaystyle e^{\ln x}=x,x>0}$

## 換基數

${\displaystyle \log _{a}M={\frac {\log _{b}M}{\log _{b}a}}}$

${\displaystyle \log _{a}M={\frac {\log M}{\log a}}}$

${\displaystyle \log _{a}M={\frac {\ln M}{\ln a}}}$

{\displaystyle {\begin{aligned}\log _{a}M&=x\\a^{x}&=M\\\log _{b}a^{x}&=\log _{b}M\\x\log _{b}a&=\log _{b}M\\x&={\frac {\log _{b}M}{\log _{b}a}}\\\log _{a}M&={\frac {\log _{b}M}{\log _{b}a}}\end{aligned}}}

## 化簡方程例子

1. ${\displaystyle \log _{9}1=x}$
2. 利用定義重寫(1)，${\displaystyle 9^{x}=1}$
3. 因此，${\displaystyle x=0}$

1. 直接利用定律，${\displaystyle \log _{a}(x^{4}y^{5})=\log _{a}(x^{4})+\log _{a}(y^{5})}$
2. 再定用多一次，${\displaystyle \log _{a}(x^{4}y^{5})=4\log _{a}x+5\log _{a}y}$

## 換基數例子

1. 因為計數機內定基數係${\displaystyle 10}$，所以要轉基數。
2. ${\displaystyle \log _{5}7={\frac {\log 7}{\log 5}}}$
3. ${\displaystyle \log _{5}7=1.209}$

## 更多例子

1. ${\displaystyle \log {\bigl (}{\frac {x^{2}-x-2}{x^{2}+3x-4}}{\bigr )}}$
2. ${\displaystyle \ln {\sqrt {\frac {x^{2}+3x-10}{x^{2}-3x+2}}}}$
3. ${\displaystyle \log _{4}19}$
4. ${\displaystyle \log _{\frac {1}{2}}5}$
5. ${\displaystyle \log _{\pi }2.7}$
6. ${\displaystyle \ln(x+1)+\ln(x-1)-2\ln(x^{2}+3)}$