课程： 3.4 日志的属性

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日志属性

Log functions are inverses of . This means the domain of one is the range of the other. This is extremely helpful when solving an equation and the unknown is in an exponent. Before solving equations, you must be able to simplify expressions containing logs. The rules of exponents are applied, but in non-obvious ways. In order to get a conceptual handle on the properties of logs, it may be helpful to continually ask, what does a log expression represent? For example, what does $\begin{aligned} \log_{10} 1, 000 \end{aligned}$ represent?
::日志函数是反向的。这意味着一个的域是另一个的域。这在解析一个方程式时非常有用, 而未知的表示是在一个引号中。在解析方程式之前, 您必须能够简化包含日志的表达方式。引用方程式的规则是适用的, 但以非明显的方式。为了获得日志属性的概念处理, 可能有必要不断询问, 日志表达式代表什么 ? 例如, log10\\\\\ 1,000 代表什么 ?

Log Properties
::日志属性

Exponential and logarithmic expressions have the same 3 components. They are each written in a different way so that a different variable is isolated. The following two equations are equivalent to one another.
::指数表达式和对数表达式有相同的3个组成部分。它们都是以不同方式写成的,因此不同的变量是孤立的。以下两个方程式是等同的。

$\begin{aligned} b^{x} = a \leftrightarrow \log_{b} a = x \end{aligned}$
::bx = a = logb =x =x = = logb =x =x

The exponential equation on the left is read “ $\begin{aligned} b \end{aligned}$ to the power $\begin{aligned} x \end{aligned}$ is $\begin{aligned} a \end{aligned}$ .” The logarithmic equation on the right is read “log base $\begin{aligned} b \end{aligned}$ of $\begin{aligned} a \end{aligned}$ is $\begin{aligned} x \end{aligned}$ ”.
::左侧的指数方程式为“b至功率x为a。” 右侧的对数方程式为“a的log b b b为x”。

The two most common bases for logs are 10 and $\begin{aligned} e \end{aligned}$ . At the PreCalculus level log by itself implies log base 10 and ln implies base $\begin{aligned} e \end{aligned}$ . ln is called the natural log . One important restriction for all log functions is that they must have strictly positive numbers in their arguments. So, if you press log -2 or log 0 on your calculator, it will give an error.
::两个最常见的日志基数是 10 和 e。在预考水平的日志本身意味着日志基数 10 和隐含的基数 e. 中称为自然日志。对所有日志函数的一个重要限制是,它们必须在其参数中有严格的正数。因此,如果在计算器上按对数 - 2 或对数 0, 就会出错。

There are three basic properties of logs that correlate to properties of exponents.
::日志有三个基本属性与指数属性相关。

Addition/Multiplication
::添加/重复

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

$\begin{aligned} b^{w + z} = b^{w} \cdot b^{z} \end{aligned}$
::体重+z=bwbz

Subtraction/Division
::减/司

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

$\begin{aligned} b^{w - z} = \frac{b^{w}}{b^{z}} \end{aligned}$
::体重 - z=bwbz

Exponentiation
::指数指数

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

$\begin{aligned} (b^{w})^{n} = b^{w \cdot n} \end{aligned}$
:bw)n=bwn

There are also a few standard results that should be memorized and should serve as baseline reference tools.
::还有一些标准结果应当记住,并应当作为基线参考工具。

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

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

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

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

Example
::示例示例示例示例

Example 1
::例1

Earlier you were asked what $\begin{aligned} \log_{10} 1, 000 \end{aligned}$ represents. A log expression represents an exponent. The expression $\begin{aligned} \log_{10} 1, 000 \end{aligned}$ represents the number 3.
::早些时候有人询问您对log101 000代表什么。日志表达式代表一个提示。表达式对101 000代表数字3。日志表达式代表一个提示。表达式对101 000代表数字3。

\begin{aligned} \log_{10} 1000 = \log_{10} 10^{3} = 3 \end{aligned}

::对数 10\\\ 1000=log10\ 103=3

The reason to keep this in mind is that it can solidify the properties of logs. For example, adding exponents implies bases are multiplied. Thus adding logs means the bases of the exponents are multiplied.
::记住这一点的原因是它能够巩固日志的属性。例如,添加引号意味着基础乘以乘以。因此添加日志意味着指数的基数乘以。

Example 2
::例2

Write the expression as a logarithm of a single argument.
::将表达式写成单个参数的对数。

$\begin{aligned} \log_{2} 12 + \log_{4} 6 - \log_{2} 24 \end{aligned}$
::log2\\\\ 12+log4\\\ 6- log2\\ 24

Note that the center expression is of a different base. First change it to base 2 by switching back to exponential form.
::注意中心表达式是不同的基数。先将其转换为基数 2 , 再转换为指数形式。

\begin{aligned} \log_{4} 6 & = x \leftrightarrow 4^{x} = 6 \\ 2^{2 x} & = 6 \leftrightarrow \log_{2} 6 = 2 x \\ x & = \frac{1}{2} \log_{2} 6 = \log_{2} 6^{\frac{1}{2}} \end{aligned}

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

Thus the expression with the same base is:
::因此,以同一基础表达的表达方式是:

\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}

::log2\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\62\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Example 3
::例3

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

Start by letting the left side of the equation be equal to $\begin{aligned} x \end{aligned}$ . Then, rewrite in exponential form, manipulate, and rewrite back in logarithmic form until you get the expression from the left side of the equation.
::以让方程式左侧等于 x 开始。然后, 以指数形式重写, 操作, 并重写以对数形式返回, 直到您从方程式左侧得到表达式。

\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, $\begin{aligned} \frac{1}{\log_{b} a} = \log_{a} b \end{aligned}$ because both expressions are equal to $\begin{aligned} x \end{aligned}$ .
::因此, 1logba=logab, 因为两个表达式都等于 x。

Example 4
::例4

Rewrite the following expression under a single log.
::在单个日志下重写以下表达式。

\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 5
::例5

True or false:
::真实或虚假 :

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

False. It is true that the log of a product is the sum of logs. It is not true that the product of logs is the log of a sum.
::假。产品的日志确实是日志的总和。日志的产物并非是数的日志。

Summary

Log functions are inverses of exponential functions.
::日志函数是指数函数的反函数。

The two most common bases for logs are 10 and e.
::日志的两个最常见的基础是10和e。

"log" by itself implies log base 10.
::“log”本身就意味着原木基数10。

"ln" implies base e, which is called the natural log.
::“在”是指基数e, 称为自然日志。

Three properties of logs are:
- $\begin{aligned} \log_{b} x + \log_{b} y = \log_{b} (x \cdot y) \end{aligned}$
  ::logbx+logby=logb(xy)
- $\begin{aligned} \log_{b} x - \log_{b} y = \log_{b} (\frac{x}{y}) \end{aligned}$
  ::logbx-logby=logb(xy)
- $\begin{aligned} \log_{b} (x^{n}) = n \cdot \log_{b} x \end{aligned}$
  ::对数b( xn) =nlogbx
::日志的三个属性是: logbx+logby=logb(xy) logbx-logby=logb(xy) logb(xy) logb(xn) logb(xn) =nlogbx

Review
::回顾

Decide whether each of the following statements is true or false. Explain.
::决定以下声明是否属实。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 under a single log and simplify.
::在单一日志下重写以下表达式并简化。

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.
::尽可能放大表达式。

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)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

章节大纲

常规

日志属性

Log Properties ::日志属性

Addition/Multiplication ::添加/重复

Subtraction/Division ::减/司

Exponentiation ::指数指数

Example ::示例示例示例示例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)