对数函数 (logarithmic function)

1. the definition of a logarithm

如果 $N=a^x(a>0,a\ne 1)$，即 $a$ 的 $x$ 次方等于 $N(a>0,a\ne 1)$，那么数 $x$ 叫做以 $a$ 为底 $N$ 的对数 (logarithm)，记作 $x={\text{log}}_a^N$。其中，$a$ 叫做对数的底数，$N$ 叫做真数，$x$ 叫做以 $a$ 为底 $N$ 的对数。

以 10 为底的对数称为常用对数，以 $e$ 为底的对数称为自然对数。

• 以 10 为底的对数称为常用对数 (common logarithm)，记为 ${\text{lg}}^N={\text{log}}_{10}^N$。
• 以无理数 $e$ ($e$ = 2.71828…) 为底的对数称为自然对数 (natural logarithm)，记为 ${\text{ln}}^N={\text{log}}_e^N$。
• 零没有对数。

$e=\underset{x\to \infty }{\lim }(1+\frac{1}{x}{)}^x=2.718281828459...$
or
$e=\underset{x\to 0}{\lim }(1+x{)}^{\frac{1}{x}}=2.718281828459...$

函数 $y={\text{log}}_a^x(a>0,a\ne 1)$ 叫做对数函数 (logarithmic function)，其中 $x$ 是自变量。$x$ 的定义域是 $(0,+\infty )$。
$y={\text{log}}_a^x$ if and only if $a^y=x$, where $x>0,b>0,$ and $b\ne 1$.

2. logarithm rules

• logarithm product rule

The logarithm of the multiplication of $x$ and $y$ is the sum of logarithm of $x$ and logarithm of $y$.
两个正数的积的对数，等于同一底数的这两个数的对数的和。

${\text{log}}_a^{x\ast y}={\text{log}}_a^x+{\text{log}}_a^y$
${\text{log}}_{10}^{3\ast 7}={\text{log}}_{10}^3+{\text{log}}_{10}^7$

• logarithm quotient rule

The logarithm of the division of $x$ and $y$ is the difference of logarithm of $x$ and logarithm of $y$.

两个正数商的对数，等于同一底数的被除数的对数减去除数对数的差。

${\text{log}}_a^{x/y}={\text{log}}_a^x-{\text{log}}_a^y$
${\text{log}}_{10}^{3/7}={\text{log}}_{10}^3-{\text{log}}_{10}^7$

• logarithm power rule

The logarithm of $x$ raised to the power of $y$ is $y$ times the logarithm of $x$.

${\text{log}}_a^{x^y}=y{\text{log}}_a^x$
${\text{log}}_{10}^{2^8}=8{\text{log}}_{10}^2$

• logarithm base change rule

The base $b$ logarithm of $x$ is base $c$ logarithm of $x$ divided by the base $c$ logarithm of $b$.

换底公式。

${\text{log}}_a^x={\text{log}}_b^x/{\text{log}}_b^a$

For example, in order to calculate ${\text{log}}_2^8$ in calculator, we need to change the base to $10$:

${\text{log}}_2^8={\text{log}}_{10}^8/{\text{log}}_{10}^2$

References

https://www.andrews.edu/~calkins/math/webtexts/numb17.htm