课程： 8.11 解决对数等

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选择节解析对数等量

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解析对数等量
"I'm thinking of a number," you tell your best friend. "The number I'm thinking of satisfies the equation $\begin{align*}\log 10x^2 - \log x = 3\end{align*}$ ." What number are you thinking of?
::“我在想一个数字,”你告诉你最好的朋友。 “我想达到公式的数值是: olog_10x2-log_x=3。”你想的号码是多少?

Solving Logarithm Equations
::解析对数等数

A logarithmic equation has the variable within the log. To solve a logarithmic equation, you will need to use the inverse property, $\begin{align*}b^{\log_b x}=x\end{align*}$ , to cancel out the log.
::对数方程式在日志中包含变量。要解析对数方程式, 您需要使用反属性( 博客bx=x) 来取消日志。

Let's solve the following logarithmic equations.
::让我们解决以下对数方程。

$\begin{align*}\log_2(x+5)=9\end{align*}$
::log2(x+5)=9

There are two different ways to solve this equation. The first is to use the definition of a logarithm.
::解析此方程式有两种不同的方法。第一个是使用对数定义。

$\begin{aligned} \log_{2} (x + 5) & = 9 \\ 2^{9} & = x + 5 \\ 512 & = x + 5 \\ 507 & = x \end{aligned}$
$\begin{align*}\log_2(x+5) &= 9 \\ 2^9 &= x+5 \\ 512 &= x+5 \\ 507 &= x\end{align*}$
::log2(x+5)=929=x+5512=x+5507=x

The second way to solve this equation is to put everything into the exponent of a 2, and then use the inverse property.
::解决这个方程式的第二个方法是把一切都放入二号的引号中然后使用反向属性

$\begin{aligned} 2^{\log_{2} (x + 5)} & = 2^{9} \\ x + 5 & = 512 \\ x & = 507 \end{aligned}$
$\begin{align*}2^{\log_2(x+5)} &= 2^9 \\ x+5 &= 512 \\ x &= 507\end{align*}$
::2log2(x+5)=29x+5=512x=507

Make sure to check your answers for logarithmic equations. There can be times when you get an extraneous solution.To check, plug in 507 for $\begin{align*}x\end{align*}$ : $\begin{align*}\log_2(507+5)=9 \rightarrow \log_2 512=9 \end{align*}$
::确保检查对数方程式的解答。您可以得到一个不相干的解答。要检查, 在 507 中插入 x: log2\\\ (507+5) = 9}log2\\\\\ 512=9 。

$\begin{align*}3 \ln(-x)-5=10\end{align*}$
::3ln( - x) - 5=10

First, add 5 to both sides and then divide by 3 to isolate the natural log.
::首先,在两边加上5,然后除以3,分离自然原木。

$\begin{aligned} 3 \ln (- x) - 5 & = 10 \\ 3 \ln (- x) & = 15 \\ \ln (- x) & = 5 \end{aligned}$
$\begin{align*}3 \ln(-x)-5 &= 10 \\ 3 \ln(-x) &= 15 \\ \ln(-x)&= 5\end{align*}$
::3ln( - x) -5=103ln( - x)=15ln( - x)=5

Recall that the inverse of the natural log is the natural number. Therefore, everything needs to be put into the exponent of $\begin{align*}e\end{align*}$ in order to get rid of the log.
::回顾自然日志的反面是自然数。因此,所有东西都需要放在e的引号中, 才能摆脱日志。

$\begin{aligned} e^{\ln (- x)} & = e^{5} \\ - x & = e^{5} \\ x & = - e^{5} \approx - 148.41 \end{aligned}$
$\begin{align*}e^{\ln(-x)} &= e^5 \\ -x &= e^5 \\ x &= -e^5 \approx -148.41\end{align*}$
::eln(-x) =e5-x=e5x=e5x* e55 148.41

Checking the answer, we have $\begin{align*}3 \ln(-(-e^5))-5=10 \rightarrow 3\ln e^5 -5 =10 \rightarrow 3 \cdot 5-5=10\end{align*}$
::在检查答案时,我们有 3ln(-(-))-5=103ln5-5=1035-5=1035-5=10

$\begin{align*}\log 5x + \log(x-1)=2\end{align*}$
::对数5x+log(x-1)=2

Condense the left-hand side using the Product Property.
::使用产品属性,使左手侧保持稳健。

$\begin{aligned} \log 5 x + \log (x - 1) = 2 \\ \log [5 x (x - 1)] = 2 \\ \log (5 x^{2} - 5 x) = 2 \end{aligned}$
$\begin{align*}\log 5x + \log (x-1)=2 \\ \log [5x(x-1)]=2 \\ \log (5x^2-5x)=2\end{align*}$
::对数5x+log(x-1)=2log[5x(x-1)]=2log(5x2-5x)=2

Now, put everything in the exponent of 10 and solve for $\begin{align*}x\end{align*}$ .
::现在,把所有的东西放在10的指数并解决x。

$\begin{aligned} 10^{\log (5 x^{2} - 5 x)} & = 10^{2} \\ 5 x^{2} - 5 x & = 100 \\ 5 x^{2} - 5 x - 100 & = 0 \\ x^{2} - x - 20 & = 0 \\ (x - 5) (x + 4) & = 0 \\ x & = 5, - 4 \end{aligned}$
$\begin{align*}10^{\log(5x^2-5x)} &= 10^2 \\ 5x^2 - 5x &= 100 \\ 5x^2 - 5x - 100 &= 0\\ x^2-x-20 &= 0 \\ (x-5)(x+4) &= 0 \\ x &=5, -4\end{align*}$
::10log( 5x2 - 5x) =1025x2 - 5x=1005x2 - 5x - 100=0x2 - x-20=0( x5) (x+4) =0x=5, - 4

Now, check both answers.
::现在,检查两个答案。

$\begin{aligned} \log 5 (5) + \log (5 - 1) & = 2 \log 5 (- 4) + \log ((- 4) - 1) = 2 \\ \log 25 + \log 4 & = 2 \log (- 20) + \log (- 5) = 2 \\ \log 100 & = 2 \end{aligned}$
$\begin{align*}\log 5(5) + \log(5-1) &= 2 \qquad \qquad \log5(-4) + \log((-4)-1)= 2 \\ \log 25 + \log 4 &= 2 \qquad \qquad \quad \ \log(-20) + \log(-5) = 2 \\ \log 100 &= 2\end{align*}$
::5(5)+log(5-1)=2log5(-4)+log((-4)-1)=2log25+log4=2log(-20)+log(-5)=2log100=2

-4 is an extraneous solution. In the step $\begin{align*}\log(-20) + \log(-5)=2\end{align*}$ , we cannot take the log of a negative number, therefore -4 is not a solution. 5 is the only solution.
::4 是一个不相干的解决办法。在步数 log( - 20) +log( - 5) = 2 中, 我们无法使用负数的日志, 因此 - 4 不是一个解决办法。 5 是唯一的解决办法。

Examples
::实例

Example 1
::例1

Earlier, you were asked to determine what number you are thinking of if the number satisfies the equation $\begin{align*}\log 10x^2 - \log x = 3\end{align*}$ .
::早些时候,有人要求您确定您在考虑的数值,如果该数值符合等式的对数 10x2-logx=3。

We can rewrite $\begin{align*}\log 10x^2 - \log x = 3\end{align*}$ as $\begin{align*}\log {\frac{10x^2}{x}} = 3\end{align*}$ and solve for x .
::我们可以重写对数10x2-logx=3 的对数10x2x=3, 并解决 x 。

$\begin{aligned} \log \frac{10 x^{2}}{x} = 3 \\ \log 10 x = 3 \\ 10^{\log 10 x} = 10^{3} \\ 10 x = 1000 \\ x = 100 \end{aligned}$
$\begin{align*}\log {\frac{10x^2}{x}} = 3\\ \log 10x = 3\\ 10^{\log10x} = 10^3\\ 10x = 1000\\ x = 100\end{align*}$
::log10x2x=3log_10x=310log_10x=10310x=1000x=100

Therefore, the number you are thinking of is 100.
::因此,你所想到的数字是100。

Example 2
::例2

Solve: $\begin{align*}9 + 2 \log_3 x=23\end{align*}$ .
::解决时间: 9+2log3x=23。

Isolate the log and put everything in the exponent of 3.
::把日志分离出来把所有东西都放进三号计数器里

$\begin{aligned} 9 + 2 \log_{3} x & = 23 \\ 2 \log_{3} x & = 14 \\ \log_{3} x & = 7 \\ x & = 3^{7} = 2187 \end{aligned}$
$\begin{align*}9 + 2 \log_3 x &= 23 \\ 2 \log_3 x &= 14 \\ \log_3 x &= 7 \\ x &= 3^7=2187\end{align*}$
::9+2log3x=232log3x=14log3x7x=37=2187

Example 3
::例3

Solve: $\begin{align*}\ln (x-1)-\ln(x+1)=8\end{align*}$ .
::溶解: In(x- 1)-ln(x+1)=8。

Condense the left-hand side using the Quotient Rule and put everything in the exponent of $\begin{align*}e\end{align*}$ .
::使用“引号规则”使左手侧松绑,将一切置于e的引文中。

$\begin{aligned} \ln (x - 1) - \ln (x + 1) & = 8 \\ \ln (\frac{x - 1}{x + 1}) & = 8 \\ \frac{x - 1}{x + 1} & = e^{8} \\ x - 1 & = (x + 1) e^{8} \\ x - 1 & = x e^{8} + e^{8} \\ x - x e^{8} & = 1 + e^{8} \\ x (1 - e^{8}) & = 1 + e^{8} \\ x & = \frac{1 + e^{8}}{1 - e^{8}} \approx - 1.0007 \end{aligned}$
$\begin{align*}\ln(x-1) - \ln(x+1) &=8 \\ \ln \left(\frac{x-1}{x+1}\right) &= 8 \\ \frac{x-1}{x+1} &= e^8 \\ x-1 &=(x+1) e^8 \\ x-1 &= x e^8 + e^8 \\ x-x e^8 &= 1 + e^8 \\ x(1- e^8) &= 1 + e^8 \\ x &= \frac{1+ e^8}{1- e^8} \approx -1.0007\end{align*}$
::In(x- 1)- ln(x+1)=8ln(x-1x+1)=8x-1x+1=8x-1=e8x-1=(x+1)-e8x- 1=xe8+e8x-xe8=1+e8x(1-e8)=1+e8x=1+e8x=1+e8x=1+e81-e8===1.007

Checking our answer, we get $\begin{align*}\ln (-1.0007-1) - \ln (-1.0007+1)=8\end{align*}$ , which does not work because you cannot take the log of a negative number . Therefore, there is no solution for this equation.
::正在检查我们的答案, 我们得到 In (- 1.0007- 1)- ln (- 1.0007+1) =8, 无效, 因为无法使用负数的日志。因此, 此等式没有解决方案。

Example 4
::例4

Solve: $\begin{align*}\frac{1}{2}\log_5(2x+5)=2\end{align*}$ .
::解决时间: 12log5( 2x+5) = 2 。

Multiply both sides by 2 and put everything in the exponent of a 5.
::将两边乘以2 然后把所有东西都放进5号的指数里

$\begin{aligned} \frac{1}{2} \log_{5} (2 x + 5) & = 2 \\ \log_{5} (2 x + 5) & = 4 \\ 5^{\log_{5} (2 x + 5)} & = 5^{4} \\ 2 x + 5 & = 625 \\ 2 x & = 620 \\ x & = 310 \end{aligned}$
$\begin{align*}\frac{1}{2} \log_5(2x+5)&= 2 \\ \log_5(2x+5)&=4 \\ {5}^{\log_{5}{(2x+5)}} &=5^4\\ 2x+5 &= 625 \\ 2x &=620 \\ x &= 310\end{align*}$
::12log5(2x+5)=2log5(2x+5)=45log5(2x+5)=542x+5=6252x=620x=310

Review
::回顾

Use properties of logarithms and a calculator to solve the following equations for $\begin{align*}x\end{align*}$ . Round answers to three decimal places and check for extraneous solutions.
::使用对数属性和计算器解析 x. 的下列方程式。对小数点后三个位数的圆形答案, 并检查外部解决方案。

$\begin{align*}\log_2 x =15\end{align*}$
::对数 2x=15

$\begin{align*}\log_{12} x = 2.5\end{align*}$
::对数 12x=2.5

$\begin{align*}\log_9 (x-5) =2\end{align*}$
::对数 9(x- 5)=2

$\begin{align*}\log_7(2x+3)=3\end{align*}$
::对数 7( 2x+3)=3

$\begin{align*}8 \ln(3-x)=5\end{align*}$
::8ln( 3- x)=5

$\begin{align*}4 \log_3 3x-\log_3 x=5\end{align*}$
::4log33x-log3x=5

$\begin{align*}\log(x+5) + \log x = \log 14\end{align*}$
:x+5)+logx=log14

$\begin{align*}2 \ln x - \ln x =0\end{align*}$
::2lnxx- lnxx=0

$\begin{align*}3 \log_3(x-5) = 3\end{align*}$
::3log3(x-5)=3

$\begin{align*}\frac{2}{3} \log_3 x=2\end{align*}$
::23log3x=2

$\begin{align*}5 \log \frac{x}{2} -3 \log \frac{1}{x} = \log 8\end{align*}$
::5log_x2 - 3log_ 1x=log_ 8

$\begin{align*}2 \ln x^{e+2} - \ln x=10\end{align*}$
::2nxe+2-lnx=10

$\begin{align*}2 \log_6 x+1 = \log_6(5x+4)\end{align*}$
::2log6_x+1=log6(5x+4)

$\begin{align*}2 \log_{\frac{1}{2}}x+2=\log_{\frac{1}{2}}(x+10)\end{align*}$
::2log12x+2=log12(x+10)

$\begin{align*}3 \log_{\frac{2}{3}} x-\log_{\frac{2}{3}} 27 = \log_{\frac{2}{3}}8\end{align*}$
::3log23\x-log23\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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

解析对数等量

Solving Logarithm Equations ::解析对数等数

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)

Solving Logarithm Equations
::解析对数等数

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)