Course: 8.9 对数权财产 | The Way To Learn

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对对数的权力属性
The hypotenuse of a right triangle has a length of $\begin{align*}\log_3 27^8\end{align*}$ . How long is the triangle's hypotenuse?
::右三角形的下限长度为 log3 278. 三角形的下限持续多久?

Power Property
::电力产权

The last property of logs is the Power Property .
::原木的最后一项财产是电力财产。

$\begin{align*}\log_b x=y\end{align*}$
::对数bx=y

Using the definition of a log, we have $\begin{align*}b^y=x\end{align*}$ . Now, raise both sides to the $\begin{align*}n\end{align*}$ power.
::使用日志的定义, 我们有 by=x 。现在, 将两边都提升到 n权力。

$\begin{aligned} (b^{y})^{n} & = x^{n} \\ b^{n y} & = x^{n} \end{aligned}$
$\begin{align*}(b^y)^n &= x^n \\ b^{ny} &= x^n\end{align*}$
:n)n=xnbny=xn)

Let’s convert this back to a log with base $\begin{align*}b\end{align*}$ , $\begin{align*}\log_b x^n=ny\end{align*}$ . Substituting for $\begin{align*}y\end{align*}$ , we have $\begin{align*}\log_b x^n=n \log_b x\end{align*}$ .
::让我们将此转换为 b, logb_xn=ny 的日志。替换 y, 我们有logb_xn=nlogb_x 。

Therefore, the Power Property says that if there is an exponent within a logarithm, we can pull it out in front of the logarithm.
::因此,电力财产公司说,如果在对数内有一个推算符,我们可以在对数前把它拉出来。

Let's u se the Power Property to expand the following logarithms.
::让我们利用电力财产来扩大以下对数。

$\begin{align*}\log_6 17x^5\end{align*}$
::对数 6\\\\\17x5

To expand this log, we need to use the Product Property and the Power Property.
::为了扩大这一日志,我们需要使用产品产权和电力产权。

$\begin{aligned} \log_{6} 17 x^{5} & = \log_{6} 17 + \log_{6} x^{5} \\ = \log_{6} 17 + 5 \log_{6} x \end{aligned}$
$\begin{align*}\log_6 17x^5 &= \log_6 17 + \log_6 x^5 \\ &= \log_6 17 + 5\log_6 x\end{align*}$
::对数 6\\\ 17x5=log6\\ 17+log6\\\ x5=log6\\ 17+5log6\ xx

$\begin{align*}\ln \left(\frac{2x}{y^3}\right)^4\end{align*}$
::In( 2xy3) 4

We will need to use all three properties to expand this problem. Because the expression within the natural log is in parenthesis, start with moving the $\begin{align*}4^{th}\end{align*}$ power to the front of the log.
::我们需要使用所有三个属性来扩大这个问题。因为自然日志中的表达方式在括号中, 开始将第四电源移到日志前端。

$\begin{aligned} \ln {(\frac{2 x}{y^{3}})}^{4} & = 4 \ln \frac{2 x}{y^{3}} \\ = 4 (\ln 2 x - \ln y^{3}) \\ = 4 (\ln 2 + \ln x - 3 \ln y) \\ = 4 \ln 2 + 4 \ln x - 12 \ln y \end{aligned}$
$\begin{align*}\ln \left(\frac{2x}{y^3}\right)^4 &= 4 \ln \frac{2x}{y^3} \\ &= 4(\ln 2x - \ln y^3)\\ &= 4(\ln 2 + \ln x - 3 \ln y) \\ &= 4 \ln2 + 4 \ln x - 12 \ln y\end{align*}$
::In( 2xy3) 4= 4ln2xy3= 4( ln2x- ln3) = 4( ln2+lnx- 3ln) = 4ln2+4ln1x-12ln

Depending on how your teacher would like your answer, you can evaluate $\begin{align*}4\ln2 \approx 2.77\end{align*}$ , making the final answer $\begin{align*}2.77 + 4\ln x - 12\ln y\end{align*}$ .
::取决于您的老师对您答案的满意程度, 您可以评估 4ln22. 77, 得出最后答案 2. 77+4lnx- 12lny 。

Now, let's condense $\begin{align*}\log 9 - 4\log 5 - 4\log x + 2\log 7 + 2\log y\end{align*}$ .
::现在,让我们来压缩对数\\\\4log\5 -4log\5\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

This is the opposite of the previous two problems . Start with the Power Property.
::这与前两个问题正好相反。从电力财产开始。

$\begin{aligned} \log 9 - 4 \log 5 - 4 \log x + 2 \log 7 + 2 \log y \\ \log 9 - \log 5^{4} - \log x^{4} + \log 7^{2} + \log y^{2} \end{aligned}$
$\begin{align*}&\log 9 - 4\log 5 - 4\log x + 2\log7 + 2\log y \\ &\log 9 - \log 5^4 - \log x^4 + \log 7^2 + \log y^2\end{align*}$
::\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\

Now, start changing things to division and multiplication within one log.
::现在,开始把事情改变成一个日志内的分割和乘法。

$\begin{align*}\log \frac{9 \cdot 7^2 y^2}{5^4 x^4}\end{align*}$
::log=% =% =% =% =% =% =% =% =% =% =% =% =% =% =% =% =% =% =% =% =% =% =% =% =% =%

Lastly, combine like terms.
::最后,将类似术语结合起来。

$\begin{align*}\log \frac{441 y^2}{625 x^4}\end{align*}$
::log441y2625x4

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the length of the triangle's hypotenuse.
::早些时候,有人要求你找到三角形的下垂长度

We can rewrite $\begin{align*}\log_3 27^8\end{align*}$ and $\begin{align*}8\log_3 27\end{align*}$ and solve.
::我们可以重写对数3278和8log327 并解决。

$\begin{aligned} 8 \log_{3} 27 \\ = 8 \cdot 3 \\ = 24 \end{aligned}$
$\begin{align*}8\log_3 27\\ = 8 \cdot 3\\ = 24\end{align*}$
::8log327=83=24

Therefore, the triangle's hypotenuse is 24 units long.
::因此三角形的下限是24单位长

Example 2
::例2

Expand the following expression: $\begin{align*}\ln x^3\end{align*}$ .
::展开以下表达式: Inx3 。

The only thing to do here is apply the Power Property: $\begin{align*}3 \ln x\end{align*}$ .
::这里唯一需要做的是应用电力财产:3lnx。

Example 3
::例3

Expand the following expression: $\begin{align*}\log_{16} \frac{x^2 y}{32 z^5}\end{align*}$ .
::展开以下表达式: log16\\ x2y32z5。

Let’s start with using the Quotient Property.
::让我们从引用属性开始。

$\begin{aligned} \log_{16} \frac{x^{2} y}{32 z^{5}} = \log_{16} x^{2} y - \log_{16} 32 z^{5} \end{aligned}$
$\begin{align*}\log_{16} \frac{x^2 y}{32 z^5} = \log_{16} x^2y - \log_{16} 32z^5\end{align*}$
::对数 16\\ x2y32z5=log16\ x2y-log16\\\32z5

Now, apply the Product Property, followed by the Power Property.
::现在,应用产品产权,然后是电力产权。

$\begin{aligned} = \log_{16} x^{2} + \log_{16} y - (\log_{16} 32 + \log_{16} z^{5}) \\ = 2 \log_{16} x + \log_{16} y - \frac{5}{4} - 5 \log_{16} z \end{aligned}$
$\begin{align*}&= \log_{16}x^2 + \log_{16} y - \left(\log_{16} 32 + \log_{16} z^5 \right) \\ &= 2 \log_{16} x + \log_{16} y - \frac{5}{4} -5 \log_{16}z\end{align*}$
::=log16_x2+log16_y_(log16_32+log16_z5)=2log16_x+log16_y_54_5log16_z

Simplify $\begin{align*}\log_{16} 32 \rightarrow 16^n = 32 \rightarrow 2^{4n} = 2^5\end{align*}$ and solve for $\begin{align*}n\end{align*}$ . Also, notice that we put parenthesis around the second log once it was expanded to ensure that the $\begin{align*}z^5\end{align*}$ would also be subtracted (because it was in the denominator of the original expression).
::简化对数 163216n=3224n=25 并解决n. 另外,请注意,在扩展第二个日志以确保Z5也减去后,我们将在第二个日志上加括号(因为它是原表达式的分母)。

Example 4
::例4

Expand the following expression: $\begin{align*}\log (5c^4)^2\end{align*}$ .
::展开以下表达式: log( 5c4) 2。

For this problem, you will need to apply the Power Property twice.
::对于这个问题,您需要两次使用电力财产。

$\begin{aligned} \log (5 c^{4})^{2} & = 2 \log 5 c^{4} \\ = 2 (\log 5 + \log c^{4}) \\ = 2 (\log 5 + 4 \log c) \\ = 2 \log 5 + 8 \log c \end{aligned}$
$\begin{align*}\log (5c^4)^2 &= 2 \log 5c^4 \\ &= 2(\log 5 + \log c^4) \\ &= 2(\log 5 + 4 \log c) \\ &= 2 \log 5 + 8 \log c\end{align*}$
::log*5c4=2log*5c4=2(log*5+log*4)=2(log*5+4log*4c)=2log*5+8log*c)=2log*5+8log*c

Important Note: You can write this particular log several different ways. Equivalent logs are: $\begin{align*}\log 25 + 8 \log c, \log 25 + \log c^8\end{align*}$ and $\begin{align*}\log 25c^8\end{align*}$ . Because of these properties, there are several different ways to write one logarithm.
::重要注意 : 您可以以几种不同的方式写入此特定日志。等值日志是: log\ 25+8logc, log\ 25+logc8 和 log\ 25c8. 由于这些属性, 有几种不同的方式来写入一个对数。

Example 5
::例5

Condense into one log: $\begin{align*}\ln 5 - 7 \ln x^4 + 2 \ln y\end{align*}$ .
::整合成一个日志 : 5 - 7ln x4+2ln y 。

To condense this expression into one log, you will need to use all three properties.
::要将这个表达式压缩成一个日志, 您需要使用所有三个属性。

$\begin{aligned} \ln 5 - 7 \ln x^{4} + 2 \ln y & = \ln 5 - \ln x^{28} + \ln y^{2} \\ = \ln \frac{5 y^{2}}{x^{28}} \end{aligned}$
$\begin{align*}\ln 5 - 7 \ln x^4 + 2 \ln y &= \ln 5 - \ln x^{28} + \ln y^2 \\ &= \ln \frac{5 y^2}{x^{28}}\end{align*}$
::5 - 7ln_ x4+2ln_ y=ln_ 5 - ln_ x28+ln_ y2=ln_ 5y2x28

Important Note: If the problem was $\begin{align*}\ln 5 - (7 \ln x^4 + 2 \ln y)\end{align*}$ , then the answer would have been $\begin{align*}\ln \frac{5}{x^{28}y^2}\end{align*}$ . But, because there are no parentheses, the $\begin{align*}y^2\end{align*}$ is in the numerator.
::重要注意: 如果问题在于 In5-( 7lnx4+2lny) , 那么答案应该是 nn5x28y2。但是, 因为没有括号, y2 在数字中。

Review
::回顾

Expand the following logarithmic expressions.
::展开以下对数表达式。

$\begin{align*}\log_7 y^2\end{align*}$
::对数 7\\\y2

$\begin{align*}\log_{12} 5z^2\end{align*}$
::对数 12\\\\5z2

$\begin{align*}\log_4 (9x)^3\end{align*}$
::对数 4( 9x) 3

$\begin{align*}\log \left(\frac{3x}{y}\right)^2\end{align*}$
::log *( 3xy) 2

$\begin{align*}\log_8 \frac{x^3 y^2}{z^4}\end{align*}$
::对数 8x3y2z4

$\begin{align*}\log_5 \left(\frac{25x^4}{y}\right)^2\end{align*}$
::对数 5( 25x4y) 2

$\begin{align*}\ln \left(\frac{6x}{y^3}\right)^{-2}\end{align*}$
:6xy3) -2

$\begin{align*}\ln \left(\frac{e^5 x^{-2}}{y^3}\right)^6\end{align*}$
::内(e5x-2y3)6

Condense the following logarithmic expressions.
::包含以下对数表达式。

$\begin{align*}6 \log x\end{align*}$
::6logx

$\begin{align*}2 \log_6 x + 5 \log_6 y\end{align*}$
::2log6x+5log6y

$\begin{align*}3(\log x - \log y)\end{align*}$
::3(logx-logy)

$\begin{align*}\frac{1}{2} \log(x+1) - 3 \log y\end{align*}$
::12log( x+1) - 3log

$\begin{align*}4 \log_2 y + \frac{1}{3} \log_2 x^3\end{align*}$
::4log2y+13log2x3

$\begin{align*}\frac{1}{5} \left[10 \log_2 (x-3) + \log_2 32 - \log_2 y \right]\end{align*}$
::15[10log2(x-3)+log232-log2

$\begin{align*}4 \left[\frac{1}{2} \log_3 y - \frac{1}{3} \log_3 x - \log_3 z \right]\end{align*}$
::4[12log3]-13log3]-x-log3z]

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

对对数的权力属性

Power Property ::电力产权

Examples ::实例

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)

Power Property
::电力产权

Examples
::实例

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)