对数函数 (logarithmic function)

电玩女神 2023-02-27 15:53 87阅读 0赞

# **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

*  logarithm base change rule

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

换底公式。

log a x = log b x / log b a \\text\{log\}\_\{a\}^\{x \} = \\text\{log\}\_\{b\}^\{x\} / \\text\{log\}\_\{b\}^\{a\} logax=logbx/logba

For example, in order to calculate  log 2 8 \\text\{log\}\_\{2\}^\{8\} log28 in calculator, we need to change the base to  10 10 10:

log 2 8 = log 10 8 / log 10 2 \\text\{log\}\_\{2\}^\{8\} = \\text\{log\}\_\{10\}^\{8\} / \\text\{log\}\_\{10\}^\{2\} log28=log108/log102

## References ##

[https://www.andrews.edu/~calkins/math/webtexts/numb17.htm][https_www.andrews.edu_calkins_math_webtexts_numb17.htm]

[https_www.andrews.edu_calkins_math_webtexts_numb17.htm]: https://www.andrews.edu/~calkins/math/webtexts/numb17.htm