对数

定义

若a^x = b,则我们设log_a^b = x,读做以a为底的b的对数为x,其中log称为对数公式,用来求取一个数a的多少次方为b

显然,根据定义得到:

log_a^1 = 0 \tag1

log_a^a = 1 \tag2

log_a^{a^x} = x \tag3

log_a^{log_a^b} = b \tag4

log_a^{M \div N} = log_a^M - log_a^N \tag5

Proof:

$Def$ $x=log_a^M,y = log_a^N,\alpha = log_a^{M \cdot N}$ $then$ :

M \div N = a^x \div a^y = a^{x-y}

两边同时取对数

log_a^{M \div N} = log_a^{a^{x-y}} = x-y = log_a^M - log_a^N

Q.E.D

log_a^{M \cdot N} = log_a^M + log_a^N \tag6

$Def$ $x=log_a^M,y = log_a^N,\alpha = log_a^{M \cdot N}$ $then$ :

M \cdot N = a^x \cdot a^y = a^{x+y}

Proof:

两边同时取对数

log_a^{M \cdot N} = log_a^{a^{x+y}} = x+y = log_a^M + log_a^N

Q.E.D

log_a^{M^n} = n \cdot log_a^M \tag7

Proof:

Def: $x = log_a^M$

\begin{aligned} a^x &= M \\ (a^x)^n &= M ^n \\ a^{x\cdot n} &= M ^n \end{aligned}

两边现时取对数

log_a^{M ^n} = log_a ^{a^{x\cdot n} } = x \cdot n = n \cdot log_a^M

Q.E.D

log_a{1 \div N} = 0 - log_a^N = -log_a^N

\log_{a}^b = \frac{\log_c^b}{\log_c^a}

证明:

设x = \log_a^b

x = \log_a^b \rightarrow a^x = b

对a^x = b 式子的两边取对数,得到

\begin{matrix} log_c^{a^x} &=& log_c^b \\ &\rightarrow& x \cdot log_c^a = log_c^b \end{matrix}

两边同时除以log_c^a,得到

\begin{matrix} x &=& \frac{log_c^b}{log_c^a} \\ \log_a^b &=& \frac{log_c^b}{log_c^a} \end{matrix}

TODO

最多分解次数为 $\lceil log_2^n \rceil$

使用数学归纳法