對數

定義

基本定義

1. ${\displaystyle y=f(x)}$${\displaystyle y=a^{x}}$
2. ${\displaystyle x}$${\displaystyle y}$交換：${\displaystyle x=a^{y}}$
3. ${\displaystyle y}$${\displaystyle y=?}$

定義對數

${\displaystyle x>0}$${\displaystyle a>0}$ 同埋 ${\displaystyle a\neq 1}$對數函數（logarithmic function）${\displaystyle f(x)=\log _{a}x}$符合：

${\displaystyle y=\log _{a}x\iff x=a^{y}}$」。

例子

• ${\displaystyle \log _{5}25=2\implies 5^{2}=25}$
• ${\displaystyle \log _{5}({\frac {1}{25}})=-2\implies 5^{-2}={\frac {1}{25}}}$
• ${\displaystyle 9={\sqrt {81}}=81^{\frac {1}{2}}\implies \log _{81}9={\frac {1}{2}}}$
• ${\displaystyle 16=4^{2}\implies \log _{4}16=2}$

求絕對值

1. 將求嘅數等於${\displaystyle x}$${\displaystyle x=\log _{3}81}$
2. 利用等價定義轉返佢做指數表示：${\displaystyle 3^{x}=81}$
3. 解方程：${\displaystyle 3^{x}=3^{4}\implies x=4}$
4. 還原答案：${\displaystyle x=4=\log _{3}81}$

{\displaystyle {\begin{aligned}x&=\log _{5}({\frac {1}{5}})\\5^{x}&={\frac {1}{5}}\\5^{x}&=5^{-1}\\x&=-1\\x=-1&=\log _{5}({\frac {1}{5}})\end{aligned}}}

畫對數

${\displaystyle y=a^{x}}$

${\displaystyle y=\log _{a}x}$

${\displaystyle y}$軸相交點係${\displaystyle (0,1)}$ ${\displaystyle x}$軸相交點係${\displaystyle (1,0)}$

Range係${\displaystyle (0,\infty )}$ Range係${\displaystyle (-\infty ,\infty )}$

對數應用

pH值

ph值公式：${\displaystyle {\textrm {pH}}=-\log _{10}[{\textrm {H}}^{+}]}$

參考資料

1. IUPAC (1997), A. D. McNaught, A. Wilkinson (編), Compendium of Chemical Terminology ("Gold Book") (第2版), Oxford: Blackwell Scientific Publications, doi:10.1351/goldbook, ISBN 978-0-9678550-9-7