Course: 8.6 对数函数的反向属性

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对数函数的反对数属性
If you continue to study mathematics into college, you may take a course called Differential Equations. There you will learn that the solution to the differential equation $\begin{aligned} y^{'} = y \end{aligned}$ is the general function $\begin{aligned} y = C e^{x} \end{aligned}$ . What is the inverse of this function?
::如果您继续上大学学习数学,您可以选修一个名为“差异等量”的课程。在那里,您会知道,区别等式yy的解决方案是通用函数y=Cex。这个函数的反义是什么?

Inverse Properties of Logarithm s
::对数的反对数属性

By the definition of a logarithm, it is the inverse of an exponent. Therefore, a logarithmic function is the inverse of an exponential function. Recall what it means to be an inverse of a function. When two inverses are composed, they equal $\begin{aligned} x \end{aligned}$ . Therefore, if $\begin{aligned} f (x) = b^{x} \end{aligned}$ and $\begin{aligned} g (x) = \log_{b} x \end{aligned}$ , then:
::根据对数定义,它是引言的反义。因此,对数函数是指数函数的反反义。回顾它的意思是一个函数的反反义。当两个反义组成时,它们等于 x。因此,如果 f(x)=bx和g(x)=logbx,那么:

$\begin{aligned} f \circ g = b^{\log_{b} x} = x \end{aligned}$ and $\begin{aligned} g \circ f = \log_{b} b^{x} = x \end{aligned}$
::fg=blogbx=x和gf=logbbx=x

These are called the Inverse Properties of Logarithms.
::这些被称为对数的反属性。

Let's solve the following problems. W e will use the Inverse Properties of Logarithms.
::让我们解决以下的问题。我们将使用对数的反属性。

Find $\begin{aligned} 10^{\log 56} \end{aligned}$ .
::寻找 10log56。

Using the first property, we see that the bases cancel each other out. $\begin{aligned} 10^{\log 56} = 56 \end{aligned}$
::使用第一种财产,我们看到基地互相取消。 10log56=56

$\begin{aligned} e^{\ln 6} \cdot e^{\ln 2} \end{aligned}$
::2

Here, $\begin{aligned} e \end{aligned}$ and the natural log cancel out and we are left with $\begin{aligned} 6 \cdot 2 = 12 \end{aligned}$ .
::在这里,e和自然日志取消,我们只剩下62=12。

Find $\begin{aligned} \log_{4} 16^{x} \end{aligned}$ .
::查找log416x。

We will use the second property here. Also, rewrite 16 as $\begin{aligned} 4^{2} \end{aligned}$ .
::我们将使用这里的第二处地产并且将16号重写为42号

$\begin{aligned} \log_{4} 16^{x} = \log_{4} (4^{2})^{x} = \log_{4} 4^{2 x} = 2 x \end{aligned}$
::log4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Find the inverse of $\begin{aligned} f (x) = 2 e^{x - 1} \end{aligned}$ .
::查找 f( x) =2ex- 1 的反义。

Change $\begin{aligned} f (x) \end{aligned}$ to $\begin{aligned} y \end{aligned}$ . Then, switch $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ .
::修改 f(x) 至 y。然后,切换 x 和 y。

$\begin{aligned} y = 2 e^{x - 1} \\ x = 2 e^{y - 1} \end{aligned}$

::y=2ex- 1x=2ey- 1

Now, we need to isolate the exponent and take the logarithm of both sides. First divide by 2.
::现在,我们需要孤立推手并取出双方的对数。第一除以2。

$\begin{aligned} \frac{x}{2} = e^{y - 1} \\ \ln (\frac{x}{2}) = \ln e^{y - 1} \end{aligned}$

::x2=ey- 1ln_( x2)=ln_ ey- 1

Recall the Inverse Properties of Logarithms from earlier in this concept. $\begin{aligned} \log_{b} b^{x} = x \end{aligned}$ ; applying this to the right side of our equation, we have $\begin{aligned} \ln e^{y - 1} = y - 1 \end{aligned}$ . Solve for $\begin{aligned} y \end{aligned}$ .
::在此概念的前面, 回想起对数的反属性。 logbbx=x; 将它应用到方程的右侧, 我们有 iney- 1=y- 1. y 解决 y 。

$\begin{aligned} \ln (\frac{x}{2}) = y - 1 \\ \ln (\frac{x}{2}) + 1 = y \end{aligned}$

::In(x2) =y - 1ln(x2)+1=y

Therefore, $\begin{aligned} \ln (\frac{x}{2}) + 1 \end{aligned}$ is the inverse of $\begin{aligned} 2 e^{y - 1} \end{aligned}$ .
::因此,In(x2)+1是 2ey-1 的反义词。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the inverse of $\begin{aligned} y = C e^{x} \end{aligned}$ .
::早些时候,你被要求找到y=Cex的反面。

Switch x and y in the function $\begin{aligned} y = C e^{x} \end{aligned}$ and then solve for y .
::在 y= Cex 函数中切换 x 和 y, 然后为 y 解答。

$\begin{aligned} x = C e^{y} \\ \frac{x}{C} = e^{y} \\ l n \frac{x}{C} = l n (e^{y}) \\ l n \frac{x}{C} = y \end{aligned}$

::x=CeyxC=eylnxC=ln(ey)lnxC=y

Therefore, the inverse of $\begin{aligned} y = C e^{x} \end{aligned}$ is $\begin{aligned} y = l n \frac{x}{C} \end{aligned}$ .
::因此,y=Cex的反义是y=lnxC。

Example 2
::例2

Simplify $\begin{aligned} 5^{\log_{5} 6 x} \end{aligned}$ .
::简化 5log5 6x 。

Using the first inverse property, the log and the base cancel out, leaving $\begin{aligned} 6 x \end{aligned}$ as the answer.
::使用第一个反向属性,日志和基础取消,留下6x作为答案。

$\begin{aligned} 5^{\log_{5} 6 x} = 6 x \end{aligned}$

::5log56x=6x

Example 3
::例3

Simplify $\begin{aligned} \log_{9} 81^{x + 2} \end{aligned}$ .
::简化对数 9\\\\\\ 81x+2 。

Using the second inverse property and changing 81 into $\begin{aligned} 9^{2} \end{aligned}$ we have:
::使用第二个反向属性,将81改为92,我们有:

$\begin{aligned} \log_{9} 81^{x + 2} & = \log_{9} 9^{2 (x + 2)} \\ = 2 (x + 2) \\ = 2 x + 4 \end{aligned}$

::对数 9\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\4

Example 4
::例4

Find the inverse of $\begin{aligned} f (x) = 4^{x + 2} - 5 \end{aligned}$ .
::查找 f( x) =4x+2 - 5 的反义。

$\begin{aligned} f (x) & = 4^{x + 2} - 5 \\ y & = 4^{x + 2} - 5 \\ x & = 4^{y + 2} - 5 \\ x + 5 & = 4^{y + 2} \\ \log_{4} (x + 5) & = y + 2 \\ \log_{4} (x + 5) - 2 & = y \end{aligned}$

::f( x) = 4x+2-2- 5y= 4x+2-2-5x= 4y+2-5x+5= 4y+2log4( x+5) =y+2log4( x+5) - 2=y

Review
::回顾

Use the Inverse Properties of Logarithms to simplify the following expressions.
::使用对数的逆属性来简化以下表达式。

$\begin{aligned} \log_{3} 27^{x} \end{aligned}$
::对数 327x

$\begin{aligned} \log_{5} {(\frac{1}{5})}^{x} \end{aligned}$
::对数 5( 15) x

$\begin{aligned} \log_{2} {(\frac{1}{32})}^{x} \end{aligned}$
::log2(132)x

$\begin{aligned} 10^{\log (x + 3)} \end{aligned}$
::10log( x+3)

$\begin{aligned} \log_{6} 36^{(x - 1)} \end{aligned}$
::对数 636(x- 1)

$\begin{aligned} 9^{\log_{9} (3 x)} \end{aligned}$
::9log9( 3x)

$\begin{aligned} e^{\ln (x - 7)} \end{aligned}$
::eln(x-7)

$\begin{aligned} \log {(\frac{1}{100})}^{3 x} \end{aligned}$
::log*( 1100)3x

$\begin{aligned} \ln e^{(5 x - 3)} \end{aligned}$
:5x-3)内

Find the inverse of each of the following exponential functions.
::查找下列指数函数的反向。

$\begin{aligned} y = 3 e^{x + 2} \end{aligned}$
::y=3ex+2 y=3ex+2

$\begin{aligned} f (x) = \frac{1}{5} e^{\frac{x}{7}} \end{aligned}$
:xx)=15ex7

$\begin{aligned} y = 2 + e^{2 x - 3} \end{aligned}$
::y=2+e2x-3

$\begin{aligned} f (x) = 7^{\frac{3}{x} + 1 - 5} \end{aligned}$
:xx)=73x+1-5)

$\begin{aligned} y = 2 (6)^{\frac{x - 5}{2}} \end{aligned}$
::y=2(6)x-52

$\begin{aligned} f (x) = \frac{1}{3} (8)^{\frac{x}{2} - 5} \end{aligned}$
:xx)=13(8)x2-5

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。

Section outline

General

对数函数的反对数属性

Inverse Properties of Logarithm s ::对数的反对数属性

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Inverse Properties of Logarithm s
::对数的反对数属性

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)