对数函数 (logarithmic function)

1. the definition of a logarithm

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

以 10 为底的对数称为常用对数，以 e e 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 lg N = log 10 N \text{lg}^{N} = \text{log}_{10}^{N} lgN=log10N。
以无理数 e e e ( e e e = 2.71828…) 为底的对数称为自然对数 (natural logarithm)，记为 ln N = log e N \text{ln}^{N} = \text{log}_{e}^{N} lnN=logeN。
零没有对数。

e = lim ⁡ x → ∞ ( 1 + 1 x ) x = 2.718281828459… e = {\underset {x \rightarrow \infty}{\operatorname {lim}}}(1 + \frac{1}{x})^{x} = 2.718281828459… e=x→∞lim(1+x1)x=2.718281828459…
or
e = lim ⁡ x → 0 ( 1 + x ) 1 x = 2.718281828459… e = {\underset {x \rightarrow 0}{\operatorname {lim}}}(1 + x)^{\frac{1}{x}} = 2.718281828459… e=x→0lim(1+x)x1=2.718281828459…

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

2. logarithm rules

logarithm product rule

The logarithm of the multiplication of x x x and y y y is the sum of logarithm of x x x and logarithm of y y 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。

log a x ∗ y = log a x + log a y \text{log}_{a}^{x * y} = \text{log}_{a}^{x} +\text{log}_{a}^{y} logax∗y=logax+logay
log 10 3 ∗ 7 = log 10 3 + log 10 7 \text{log}_{10}^{3 * 7} = \text{log}_{10}^{3} +\text{log}_{10}^{7} log103∗7=log103+log107

logarithm quotient rule

The logarithm of the division of x x x and y y y is the difference of logarithm of x x x and logarithm of y y 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。

log a x / y = log a x − log a y \text{log}_{a}^{x / y} = \text{log}_{a}^{x} - \text{log}_{a}^{y} logax/y=logax−logay
log 10 3 / 7 = log 10 3 − log 10 7 \text{log}_{10}^{3 / 7} = \text{log}_{10}^{3} - \text{log}_{10}^{7} log103/7=log103−log107

logarithm power rule

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

log a x y = y log a x \text{log}_{a}^{x^y} = y \text{log}_{a}^{x} logaxy=ylogax
log 10 2 8 = 8 log 10 2 \text{log}_{10}^{2^8} = 8 \text{log}_{10}^{2} log1028=8log102