节: 日志属性 | 4.4 日志属性

章节大纲

Introduction
::导言

Consider the exponential function

\begin{aligned} y = 2^{x} . \end{aligned}

Its inverse is

\begin{aligned} y = \log_{2} x . \end{aligned}

To progress further with this class of functions, we need to clarify what a log expression represents. Consider the following:

Function Domain: $\begin{aligned} x \end{aligned}$	Function Range: $\begin{aligned} y = 2^{x} \end{aligned}$	Inverse Domain: $\begin{aligned} y = 2^{x} \end{aligned}$	Inverse Range: $\begin{aligned} x \end{aligned}$
-2	$\begin{aligned} 2^{- 2} = \frac{1}{4} \end{aligned}$	$\begin{aligned} 2^{- 2} = \frac{1}{4} \end{aligned}$	-2
-1	$\begin{aligned} 2^{- 1} = \frac{1}{2} \end{aligned}$	$\begin{aligned} 2^{- 1} = \frac{1}{2} \end{aligned}$	-1
0	$\begin{aligned} 2^{0} = 1 \end{aligned}$	$\begin{aligned} 2^{0} = 1 \end{aligned}$	0
1	$\begin{aligned} 2^{1} = 2 \end{aligned}$	$\begin{aligned} 2^{1} = 2 \end{aligned}$	1
2	$\begin{aligned} 2^{2} = 4 \end{aligned}$	$\begin{aligned} 2^{2} = 4 \end{aligned}$	2

Using the columns regarding the inverse function, we can rewrite the table as follows:
::使用反向函数的列,我们可以将表格改写如下:

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = \log_{2} x \end{aligned}$
$\begin{aligned} \frac{1}{4} \end{aligned}$	$\begin{aligned} \log_{2} \frac{1}{4} = - 2 \end{aligned}$
$\begin{aligned} \frac{1}{2} \end{aligned}$	$\begin{aligned} \log_{2} \frac{1}{2} = - 1 \end{aligned}$
1	$\begin{aligned} \log_{2} 1 = 0 \end{aligned}$
2	$\begin{aligned} \log_{2} 2 = 1 \end{aligned}$
4	$\begin{aligned} \log_{2} 4 = 2 \end{aligned}$

This table illustrates the meaning of the logarithm. The logarithm is an exponent. The expression $\begin{aligned} \log_{2} 4 = 2 \end{aligned}$ can be read: "The power (log) to which 2 is taken (subscript) to get 4 (argument) is 2." Practice reading the right column of the last table using that language.
::此表格显示了对数的含义。对数是一个引号。表达式 log2\\\\ 4= 2 可以读作 : “ 使用 2 获得 4 (参数) 的功率( log) 是 2 。练习使用该语言读取最后一个表格右栏。

Logarithms
::对数对数

The definition of a logarithm can be expressed in following two equivalent equations:
::对数的定义可以用以下两个等量方程式表示:

$\begin{aligned} a = b^{x} ⟺ x = \log_{b} a . \end{aligned}$

The exponential equation is read " $\begin{aligned} b \end{aligned}$ to the $\begin{aligned} x \end{aligned}$ is $\begin{aligned} a \end{aligned}$ ." The logarithmic equation is read "log base $\begin{aligned} b \end{aligned}$ of $\begin{aligned} a \end{aligned}$ is $\begin{aligned} x \end{aligned}$ ."
::指数方程式读为“ B 到 x 是 a 。”对数方程式读为“ log b b b a is x 。 ”

Properties of Logarithms
::对数属性的对数属性

There are several properties of logarithms that are used as tools to solve equations with exponential and logarithmic expressions. Each of these can be easily established by translating the expression into words.
::对数有几个属性, 用作用指数表达式和对数表达式解析方程式的工具。将表达式转换成单词可以很容易地确定其中的每个属性。

Properties of Logarithms
::对数属性的对数属性

For $\begin{aligned} b > 0 \end{aligned}$ ,
::b>0,

$\begin{aligned} \log_{b} 1 = 0 \end{aligned}$
::对数b=0

Verification: Rewrite the equation using the definition $\begin{aligned} b^{0} = 1, \end{aligned}$ which is a more familiar statement.
::校验: 使用 b0=1 定义重写方程式, 即更熟悉的语句。

$\begin{aligned} \log_{b} b = 1 \end{aligned}$
::对数bb=1

Verification: Rewrite the equation using the definition $\begin{aligned} b^{1} = b \end{aligned}$ , which is a more familiar statement.
::校验:使用定义b1=b重写方程式,这是更熟悉的语句。

$\begin{aligned} \log_{b} (b^{x}) = x \end{aligned}$
::对数b(bx)=x

Verification: The equation can be rewritten using the definition $\begin{aligned} b^{x} = b^{x} \end{aligned}$ .
Consider the function $\begin{aligned} f (x) = \log_{b} x \end{aligned}$ . Then the inverse function is $\begin{aligned} f^{- 1} (x) = b^{x} \end{aligned}$ . Since the two functions are inverses, $\begin{aligned} (f \circ f^{- 1}) (x) = f (f^{- 1} (x)) = \log_{b} f^{- 1} (x) = \log_{b} b^{x} . \end{aligned}$ Recall that for , $\begin{aligned} (f \circ f^{- 1}) (x) = x \end{aligned}$ . Thus, $\begin{aligned} \log_{b} b^{x} = x \end{aligned}$ .
::校验 : 方程式可以使用定义 bx=bx 重写。考虑函数 f( x) = logb = x。然后反函数为 f- 1 (x) = bx 。因为这两个函数是反函数, (fäff- 1 (x) = f(f- 1 (x)) = logb- 1 (x) = logb= logbx 。 recall that for , (fäf- 1 (x) =x) =x。因此, logb_ bx=x 。

$\begin{aligned} b^{\log_{b} x} = x, x > 0 \end{aligned}$
::博客bx=x, x>0

Verification: The inverse function argument works the same way as the previous property, $\begin{aligned} (f^{- 1} \circ f) (x) = b^{\log_{b} x} = x \end{aligned}$ .
::校验: 反函数参数与先前的属性( f- 1f (x) =blogbx=x) 相同。

Along with the properties of logarithms, there are rules for logarithms and exponents that detail how to handle arithmetic operators.
::除了对数的属性外,还有对数和引号的规则,详细规定了如何处理算术操作员。

Rules of Logarithms & Exponents
::对数和指数规则

For $\begin{aligned} b, x, and y > 0 \end{aligned}$ :
::对于 b、 x 和 y>0 :

Addition or Multiplication of Logarithms
::对数的加法或乘法的乘法

Logarithmic form: $\begin{aligned} \log_{b} (x \cdot y) = \log_{b} x + \log_{b} y \end{aligned}$
::对数表: logb( xy) =logbx+logby

Exponential form: $\begin{aligned} b^{w} \cdot b^{z} = b^{w + z} \end{aligned}$
::指数表:bwz=bw+z

Subtraction or D ivision of Logarithms
::减法或对数分

Logarithmic form: $\begin{aligned} \log_{b} (\frac{x}{y}) = \log_{b} x - \log_{b} y \end{aligned}$
::对数表:logb(xy)=logbx-logby

Exponential form: $\begin{aligned} \frac{b^{w}}{b^{z}} = b^{w - z} \end{aligned}$
::指数形式: 体重/bz=bw-z

Power to a Power
::权力对一权力国的权力

Logarithmic form: $\begin{aligned} \log_{b} (x^{n}) = n \cdot \log_{b} x \end{aligned}$
::对数表: logb( xn) =nlogbx

Exponential form: $\begin{aligned} (b^{w})^{n} = b^{w \cdot n} \end{aligned}$
::指数形式bw)n=bwn

Change of Base
::基数变动

$\begin{aligned} \log_{b} x = \frac{\log_{a} x}{\log_{a} b} \end{aligned}$
::对数bx=logaxlogab

$\begin{aligned} \log_{\frac{1}{b}} x = - \log_{b} x \end{aligned}$
::对数1bxlogbx

$\begin{aligned} \log_{a^{n}} x = \frac{1}{n} \log_{a} x \end{aligned}$
::logan=1nloga=1nloga=x logan=1nloga=x

Examples
::实例

Example 1
::例1

Simplify the following expressions:
::简化以下表达式:

a) $\begin{aligned} \log_{4} 64 \end{aligned}$
::a) log464

Solution:
::解决方案 :

$\begin{aligned} \log_{4} 64 = \log_{4} 4^{3} = 3 \cdot \log_{4} 4 = 3 \cdot 1 = 3 \end{aligned}$

::log4=3=3=3=3=1=3=3=1=3=4=4=4=1=1=3=4=4=4=4=4=3=1=3=3=3=3=4=4=4=4=4=4=4=4=1=1=1=3=3=3=3=3=3=4=3=4=4=4=4=4=4=4=4=4=4=4=4=4=3=1=1=1=3=3=3=3=3=3=3=3=3=4=4=4=4=4=4=4=4=4=4=4=4=4=3=1=1=3=3=3=3=3=3=3=3=3=3

b) $\begin{aligned} \log_{\frac{1}{2}} 32 \end{aligned}$
::b) log12 32

Solution:
::解决方案 :

$\begin{aligned} \log_{\frac{1}{2}} 32 \\ - \log_{2} 32 \\ - 5 \end{aligned}$

::对数 12\\\\\32 -log2\\\\\\\\32 -5

c) $\begin{aligned} \log_{3} 3^{5} \end{aligned}$
::c) 日数335

Solution:
::解决方案 :

$\begin{aligned} \log_{3} 3^{5} = 5 \cdot \log_{3} 3 = 5 \end{aligned}$

::对数 3\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

d) $\begin{aligned} \log_{2} 128 \end{aligned}$
::d) log2128

Solution :
::解决方案 :

$\begin{aligned} \log_{2} 128 = \log_{2} 2^{7} = 7 \end{aligned}$
::log2128=log227=7

Example 2
::例2

Write the expression as a single logarithm:
::将表达式写成单对数 :

\begin{aligned} \log_{2} 12 + \log_{2} 6^{\frac{1}{2}} - \log_{2} 24 \end{aligned}

::log2\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\24

Solution :
::解决方案 :

T he expression with the same base is
::同一基底的表达式为

\begin{aligned} \log_{2} 12 + \log_{2} 6^{\frac{1}{2}} - \log_{2} 24 & = \log_{2} (\frac{12 \cdot \sqrt{6}}{24}) \\ = \log_{2} (\frac{\sqrt{6}}{2}) . \end{aligned}

Example 3
::例3

Simplify the expression $\begin{aligned} 2 \log_{12} 144^{- 4} \end{aligned}$ using properties of logarithms.
::使用对数属性简化表达式 2log12144-4 。

Solution :
$\begin{aligned} 2 \log_{12} 144^{- 4} = - 8 \cdot \log_{12} 12^{2} = - 16 \cdot \log_{12} 12 = - 16 (1) = - 16 \end{aligned}$
::解决之道:2log12_144_4_8_8_log12_122_16_log12_12_16_16_1_1_1_16_16_

Example 4
::例4

Prove the following log identity:
::证明下列日志身份:

$\begin{aligned} \log_{a} b = \frac{1}{\log_{b} a} \end{aligned}$
::对数ab=1logba

Solution:

::解决方案 :

Let the left side of the equation be equal to $\begin{aligned} x \end{aligned}$ . R ewrite in exponential form in order to manipulate the equation to solve for $\begin{aligned} a \end{aligned}$ . Then r ewrite back in logarithmic form until you get the expression from the left side of the equation.
::让方程的左侧等于 x。重写指数形式, 以便操纵方程解答一个方程。然后重写以对数形式返回, 直到您从方程的左侧得到表达式。

\begin{aligned} \log_{a} b & = x \\ a^{x} & = b \\ a & = b^{\frac{1}{x}} \\ \log_{b} a & = \log_{b} b^{\frac{1}{x}} = \frac{1}{x} \\ x & = \frac{1}{\log_{b} a} \end{aligned}

::loga=xax=ba=b1xlogb=a=logb=b1xx=1xx=1logb=a

Therefore, $\begin{aligned} \frac{1}{\log_{b} a} = \log_{a} b \end{aligned}$ .
::因此,1logba=logab。

Example 5
::例5

Rewrite the following expression as a single logarithm:
::重写以下表达式为单对数 :

$\begin{aligned} \ln e - \ln 4 x + 2 (e^{\ln x} \cdot \ln 5) \end{aligned}$ .
::===================================================================================================================================================== ============================================================================================================================ ==========================================================================================================================================================================================================================================

Solution :
::解决方案 :

\begin{aligned} \ln e - \ln 4 x + 2 (e^{\ln x} \cdot \ln 5) & = \ln (\frac{e}{4 x}) + 2 x \cdot \ln 5 \\ = \ln (\frac{e}{4 x}) + \ln (5^{2 x}) \\ = \ln (\frac{e \cdot 5^{2 x}}{4 x}) \end{aligned}

::内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内二内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内内

Example 6
::例6

True or false?
::真实还是虚假?

$\begin{aligned} (\log_{3} 4 x) \cdot (\log_{3} 5 y) = \log_{3} (4 x + 5 y) \end{aligned}$
:log34x) (log35y) = log3(4x+5y)

Solution:

::解决方案 :

False! The logarithm of a product is the sum of logs. However, it is not true that the product of logs is the log of a sum.
::假的!一个产品的对数是原木的和数,但是,原木的对数不是总数的。

Example 7
::例7

Simplify $\begin{aligned} \log_{2} 48 - \log_{4} 36 \end{aligned}$ .
::简化对数248-log436。

Solution:
::解决方案 :

U se the formula when one of the bases is an exponent of the other :
::当一个基点是另一个基点的引号时使用公式:

\begin{aligned} \log_{2} 48 - \log_{4} 36 & = \frac{\log_{4} 48}{\log_{4} 2} - \log_{4} 36 \\ = \frac{\log_{4} 48}{\log_{4} 4^{\frac{1}{2}}} - \log_{4} 36 \\ = \frac{\log_{4} 48}{\frac{1}{2}} - \log_{4} 36 \\ = 2 \log_{4} 48 - \log_{4} 36 \\ = \log_{4} 48^{2} - \log_{4} 36 \\ = \log_{4} \frac{48^{2}}{36} \\ = \log_{4} 64 \\ = 3 \end{aligned}

::log2\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Summary
::摘要

Definition of logarithm: $\begin{aligned} b^{x} = a \leftrightarrow \log_{b} a = x \end{aligned}$
::对数定义: bx=alogba=x

Properties:
1. $\begin{aligned} \log_{b} 1 = 0 \end{aligned}$
  ::对数b=0
2. $\begin{aligned} \log_{b} b = 1 \end{aligned}$
  ::对数bb=1
3. $\begin{aligned} \log_{b} (b^{x}) = x \end{aligned}$
  ::对数b(bx)=x
4. $\begin{aligned} b^{\log_{b} x} = x \end{aligned}$
  ::博客bx=x
::属性: logb1=0 logbb=1 logb(bx)=x 博客bx=x

Rules:
1. $\begin{aligned} \log_{b} (x \cdot y) = \log_{b} x + \log_{b} y \end{aligned}$
  ::logb(xy) =logbx+logby
2. $\begin{aligned} \log_{b} (\frac{x}{y}) = \log_{b} x - \log_{b} y \end{aligned}$
  ::logb(xy) =logbx-logby
3. $\begin{aligned} \log_{b} (x^{n}) = \log_{b} (x)^{n} = n \cdot \log_{b} x \end{aligned}$
  ::logb(xn) =logb(x)n =nlogbx
4. $\begin{aligned} \log_{b} x = \frac{\log_{a} x}{\log_{a} b} \end{aligned}$
  ::对数bx=logaxlogab
5. $\begin{aligned} \log_{\frac{1}{b}} x = - \log_{b} x \end{aligned}$
  ::对数1bxlogbx
6. $\begin{aligned} \log_{a^{n}} x = \frac{1}{n} \log_{a} x \end{aligned}$
  ::logan=1nloga=1nloga=x logan=1nloga=x
::规则 : logb(xy) = logb(xy) = logb(xy) = logb(x}) = logb(xx) = logb(x) x logb= loga}xloga(xy) àb logb log1bx logb*x loganx=1nlogax

Review
::回顾

Decide whether each of the statements below is true or false. Explain.
::决定下面的每个声明是真实的还是虚假的,请解释。

1. $\begin{aligned} \frac{\log x}{\log y} = \log (\frac{x}{y}) \end{aligned}$
::1. 对数xlogy=log(xy)

2. $\begin{aligned} (\log x)^{n} = n \log x \end{aligned}$
::2. (logx)n=nlogx

3. $\begin{aligned} \log x + \log y = \log x y \end{aligned}$
::3. 对数x+logy=logxy

Rewrite each of the following expressions as a single logarithm:
::将下列表达式中的每一表达式重写为单对数 :

4. $\begin{aligned} \log 4 x + \log (2 x + 4) \end{aligned}$
::4. 对数4x+log(2x+4)

5. $\begin{aligned} 5 \log x + \log x \end{aligned}$
::5. 5logx+logx

6. $\begin{aligned} 4 \log_{2} x + \frac{1}{2} \log_{2} 9 - \log_{2} y \end{aligned}$
::6. 4log2x+12log29-log2y

7. $\begin{aligned} 6 \log_{3} z^{2} + \frac{1}{4} \log_{3} y^{8} - 2 \log_{3} z^{4} y \end{aligned}$
::7. 6log3z2+14log3y8-2log3z4y

Expand the expression as much as possible:
::尽可能扩展表达式 :

8. $\begin{aligned} \log_{4} (\frac{2 x^{3}}{5}) \end{aligned}$
::8. log4(2x35)

9. $\begin{aligned} \ln (\frac{4 x y^{2}}{15}) \end{aligned}$
::9.(4xy215)

10. $\begin{aligned} \log (\frac{x^{2} (y z)^{3}}{3}) \end{aligned}$
::10. log(x2(yz)33)

Translate from exponential form to logarithmic form:
::从指数形式转换为对数形式:

11. $\begin{aligned} 2^{x + 1} + 4 = 14 \end{aligned}$
::11. 2x+1+4=14

Translate from logarithmic form to exponential form:
::从对数形式转换为指数形式:

12. $\begin{aligned} \log_{2} (x - 1) = 12 \end{aligned}$
::12. log2(x- 1)=12

Prove the following properties of logarithms:
::证明对数的下列属性:

13. $\begin{aligned} \log_{b^{n}} x = \frac{1}{n} \log_{b} x \end{aligned}$
::13. logbnx=1nlogbx

14. $\begin{aligned} \log_{b^{n}} x^{n} = \log_{b} x \end{aligned}$
::14. logbnxn=logbx

15. $\begin{aligned} \log_{\frac{1}{b}} \frac{1}{x} = \log_{b} x \end{aligned}$
::15. log1b1x=logbx

Review ( Answers)
::回顾(答复)

Please see the Appendix.
::请参看附录。

章节大纲

Introduction ::导言

Logarithms ::对数对数

Properties of Logarithms ::对数属性的对数属性

Properties of Logarithms ::对数属性的对数属性

Rules of Logarithms & Exponents ::对数和指数规则

Examples ::实例

Example 4 ::例4

Summary ::摘要

Review ::回顾

Review ( Answers) ::回顾(答复)

Introduction
::导言

Logarithms
::对数对数

Properties of Logarithms
::对数属性的对数属性

Properties of Logarithms
::对数属性的对数属性

Rules of Logarithms & Exponents
::对数和指数规则

Examples
::实例

Example 4
::例4

Summary
::摘要

Review
::回顾

Review ( Answers)
::回顾(答复)