节: 对数属性的对数属性 | 5.7 对数属性-interactive

章节大纲

Lesson Objectives
::经验教训目标

Apply the logarithm of a product property to write .
::应用产品属性的对数写入。

Solve equations using mu lt iple natural logarithmic properties.
::使用多个自然对数特性解析方程式。

Introduction: Seismic Activity Continued
::导言:地震活动继续

In Transformations of Log Functions, the formula $\begin{align*}M(I) = \log{\left(\tfrac{A}{{A}_{0}}\right)}\end{align*}$ was used to find the magnitude of an earthquake. The $\begin{align*}{A}_{0}\end{align*}$ accounts for the distance from the epicenter. Assuming that you were at the epicenter, this equation could be rewritten as $\begin{align*}M(I) = \log{\left(A\right)}.\end{align*}$ Suppose you want to write a formula that compares two earthquakes at their epicenters based on their magnitudes and intensities. Normally, to compare two quantities, quantity 1 would be divided by quantity 2, forming a ratio. However, i f you were to do this with a magnitude 3.0 earthquake and a magnitude 6.0 earthquake, it would appear that a magnitude 6.0 earthquake is twice as powerful. You know from the Transformations of Log Functions that each whole number on the scale is 10 times more powerful than the previous whole number. A 6.0 earthquake is actually 1,000 times more powerful than a 3.0 earthquake the same way that a kilometer is 1,000 times longer than a meter. This problem would be further complicated if you wanted to find how much more powerful a 7.3 earthquake is than a 7.1 earthquake. Understanding will help to figure this problem out.
::在日志函数的转换中, 公式M( I) =log( AA0) 用于查找地震的大小。公式M( I) =log( AA0) 用于查找地震的大小。如果A0 表示震中距离的距离。假设你在震中, 这个公式可以重写为 M( I) =log( A) 。如果您想要根据地震的大小和强度来比较两次震中地震的大小和强度, 则该公式可以重写为M( I) =log( A)。通常, 要比较两个数量, 数量1 将除以数量 2 = 2 , 形成一个比率。但是, 如果您想要在3. 0 级地震和6. 0 级地震中这样做, 则显示6. 0 级地震的强度是它的两倍。您从日志函数的变换中知道, 规模中每个整数值的功率是前一个整数的10倍。 6.0 地震的功率实际上比3. 0 强1 000倍, 其功率是比一米的1000倍。。如果您想要发现7. 1 71 地震的功率会进一步复杂化。。。

Activity 1: Logarithm of a Product Property
::活动1:产品属性的对数

Remember that l ogarithms can be viewed as inverses of and that a logarithm represents the exponent in an exponential equation .
::记住对数可以被视为对数的反比,对数代表指数方程式中的表征。

Additionally, recall t he Law of Exponents for products with the same base: $\begin{align*}b^m \cdot b^n=b^{m+n}.\end{align*}$ Use what you know about the relationships between exponential and logarithmic equations to rewrite this property for logarithms.
::此外, 提醒您注意同一基数为 bmbn=bm+n 的产品中的指数定律。使用您所知道的指数式和对数方程之间的关系来重写此属性的对数。

Since logarithms are being used, you will need to know what $\begin{align*}b^m \text{ and } b^n\end{align*}$ equal, respectively. C hoose the variables $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ to be written in logarithm form.
::由于正在使用对数, 您需要分别知道 bm 和 bn 等值。选择要以对数格式写入的变量 x 和 y 。

$\begin{align*}b^m=x\end{align*}$
::bm=x bm=x

$\begin{align*}b^n=y\end{align*}$
::bn=y

Replacing $\begin{align*}b^m \text{ and } b^n\end{align*}$ with $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ will result in the following:
::以 x 和 y 替换 bm 和 bn 将产生以下结果:

$\begin{align*}xy = {b}^{n+m}\end{align*}$
::xy=bn+m 时

Additionally, r eplace the $\begin{align*}m\end{align*}$ and $\begin{align*}n\end{align*}$ on the right side to put them in terms of $\begin{align*}x\end{align*}$ and $\begin{align*}y.\end{align*}$ To figure out what $\begin{align*}m\end{align*}$ and $\begin{align*}n\end{align*}$ equal, you will need a logarithm.
::此外,将右侧的 m 和 n 替换为 x 和 y。要确定 m 和 n 等值, 您需要一个对数。

$\begin{align*}b^m=x~ \rightarrow~ m = \log_{b}{x}\end{align*}$
::bm=x m=logbx

$\begin{align*}b^n=y ~\rightarrow~ n = \log_{b}{y}\end{align*}$
::bn=y n=logby = n=logby = bn=y n=logby

Substituting these logarithms into the equation above will give the following:
::将这些对数替换为上述方程将产生以下结果:

$\begin{align*}xy = {b}^{\log_{b}{x}+\log_{b}{y}}\end{align*}$
::xy=博客bx+logby

Taking the log of both sides will allow you to write this without the exponent.
::取下双方的日志将允许你写这个没有推手。

$\begin{align*}\log_{b}{xy} = \log_{b}{{b}^{\log_{b}{x}+\log_{b}{y}}}\end{align*}$
::logblogbx+logby =logblogbbx+logby logbx =logbx+logbx+logby logbx+logbx+logby logbx+logbx+logby logbx+logbx+logby logbx+logby logbx+logbx+logby logbx+logbx+logby logbx+logby logbx+logby logbx+logbx+logby logbx+logby logbx+logbx+logbx+logby logbx+ logbx+ logbx+logby log logbx+ logbx+ logbx+ logby logb log log logbx+ logbx+ logb logb logb logbx+ logbx+ logby log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log log logb log log log log log log log log log log log log log log log log log log log log log log log

Recall from Logarithms that log base $\begin{align*}b\end{align*}$ of $\begin{align*}b\end{align*}$ cancels because $\begin{align*}b^1=b.\end{align*}$ This step gives results in the logarithm of a product property:
::从对数返回b的对数 b的对数基准 b 取消,因为 b1=b。此步骤会得出产品属性的对数结果 :

The Logarithm of a Product Property
::产品属性的对数

$\begin{align*}\log_{b}{(xy)} = \log_{b}{x}+\log_{b}{y}\end{align*}$

We can use this property to derive the other log properties. Answer the questions below to practice using this property and derive the logarithm of a quotient property.
::我们可以使用此属性来获取其它日志属性。回答下面的问题, 以便使用此属性来练习, 并得出一个商数属性的对数。

Extension: Log Triangles
::扩展名: 日志三角

Use the interactive below to explore the logarithm of a product property further.
::使用以下互动方式进一步探讨产品属性的对数。

Activity 2: The Logarithm of a Power Property
::活动2:电力产权的对数

When taking the logarithm of a term with an exponent, a very useful property is encountered:
::在使用术语的对数与引号时,遇到一个非常有用的属性:

$\begin{align*}\log_{b}{(x^a)}=y\end{align*}$
::logb(xa)=y

To get another perspective, this can be written it in exponential form as follows:
::为了从另一个角度看问题,可以以指数形式写成如下:

$\begin{align*}\log_{b}{(x^a)}=y \rightarrow b^y=x^a\end{align*}$
::logb(xa) =yby=xa

Answer the questions below to derive the logarithm of a power property.
::回答下面的问题,得出权力财产的对数。

Discussion Question : Write the logarithm $\begin{align*}\log_{2}{\left(4^5\right)}=x\end{align*}$ in exponential form . How does this relate to $\begin{align*}5\log_{2}{4}=x?\end{align*}$
::讨论问题: 以指数形式写入对数log2( 45) =x。这与 5log24 =x 有何关系?

Activity 3: The Logarithm of a Quotient Property
::活动3:引数属性的对数

In the previous activity, the logarithm of a product property was used to derive the logarithm of a quotient property.
::在先前的活动中,产品财产的对数被用来得出商数财产的对数。

The Logarithm of a Quotient Property
::引用属性的对数

$\begin{align*}\log_{b}{\left(\frac{x}{y}\right)} = \log_{b}{x} - \log_{b}{y}\end{align*}$

You will need this property to answer the question in the introduction and compare the magnitude of the two earthquakes below.
::您需要这块地契来解答导言中的问题, 并比较以下两次地震的规模。

Earthquake 1: Magnitude of 7.3 , Intensity = $\begin{align*}{I}_{1}\end{align*}$
::地震1:7.3,强度=I1

Earthquake 2: Magnitude of 7.1, Intensity = $\begin{align*}{I}_{2}\end{align*}$
::地震2:7.1的磁度,强度=I2

Although you can't find the ratio of the magnitudes because they are scaled logarithmically, you can find the ratio of the intensities.
::虽然无法找到星等的比, 因为它们是按对数缩放的, 但能找到强度的比。

$\begin{align*}\frac{{I}_{1}}{{I}_{2}}\end{align*}$
::I1I2

However, since only the magnitudes are known, you will need to relate this ratio to the base 10 logarithm used to get the magnitudes.
::然而,由于只知道这些数值,所以您需要将这个比率与用于获取这些数值的10对数基数对数联系起来。

$\begin{align*}\log_{10}{\left(\frac{{I}_{1}}{{I}_{2}}\right)}\end{align*}$
::log10( I1I2) log10( I1I2)

U se the logarithm of a quotient property to rewrite this:
::使用商数属性的对数重写此内容 :

$\begin{align*}\log_{10}{\left(\frac{{I}_{1}}{{I}_{2}}\right)}=\log_{10}{({I}_{1})} - \log_{10}{({I}_{2})}\end{align*}$
::log10( I1I2) =log10( I1) -log10( I2)

Notice that this gives you the difference of $\begin{align*}\log_{10}{{I}_{1}} \text{ and } \log_{10}{{I}_{2}}.\end{align*}$ What are $\begin{align*}\log_{10}{{I}_{1}} \text{ and } \log_{10}{{I}_{2}}?\end{align*}$ You saw in the introduction that $\begin{align*}M = \log{\left({I}\right)}.\end{align*}$
::请注意, 这给了您log10I1和log10I2的差数。什么是log10I1和log10I2? 您在导言中看到 M=log( I) 。

•   $\begin{align*}\log_{10}{{I}_{1}} = {M}_{1}\end{align*}$
::• log10I1=M1

•   $\begin{align*}\log_{10}{{I}_{2}} = {M}_{2}\end{align*}$
::• log10I2=M2

S ubstitute the magnitudes into the formula to produce the following:
::以数值替代公式,产生以下结果:

$\begin{align*}log_{10}{\left(\frac{{I}_{1}}{{I}_{2}}\right)}={M}_{1}-{M}_{2}\end{align*}$
::log10 (I1I2) = M1 - M2

Knowing that $\begin{align*}{M}_{1} = 7.3 \text{ and } {M}_{2} = 7.1\end{align*}$ w ill produce the following:
::明知M1=7.3和M2=7.1将产生以下结果:

$\begin{align*}log_{10}{\left(\frac{{I}_{1}}{{I}_{2}}\right)}&=7.3-7.1\\\\ log_{10}{\left(\frac{{I}_{1}}{{I}_{2}}\right)}&=0.2\end{align*}$
::log10(I1I2)=7.3-7.1log10(I1I2)=0.2

Writing this in exponential form gives the following:
::以指数形式写成的该表示如下:

$\begin{align*}\frac{{I}_{1}}{{I}_{2}} &= {10}^{0.2}\\\\ \frac{{I}_{1}}{{I}_{2}} &\approx 1.584893 \end{align*}$
::I1I2 = 100.2I1I2 = 1.584893

Answer: A 7.3 magnitude earthquake is approximately 1.58 times more powerful than a 7.1 magnitude earthquake.
::答复:7.3级地震比7.1级地震强约1.58倍。

A 6.0 earthquake is 1,000 times more powerful than a 3 .0 earthquake. Use the formula to verify this. Once you have verified the formula, answer the questions below.
::6.0地震的威力是3.0地震的1000倍。使用公式来验证这一点。一旦您验证了公式, 请回答下面的问题。

Activity 4: Connecting to Natural Log
::活动4:与自然日志连接

Since the natural log is just a logarithm with base $\begin{align*}e,\end{align*}$ natural logarithms have all the same properties as logarithms.
::由于自然对数只是与基数e的对数,因此自然对数具有与对数相同的特性。

Example
::示例示例示例示例

Using the formula for the growth of a population $\begin{align*}P={P}_{0}\cdot{e}^{rt},\end{align*}$ where $\begin{align*}{P}_{0}\end{align*}$ is 7.6 billion people, and $\begin{align*}r\end{align*}$ is 1.1%, how long will it take for the population to reach 10 billion people?
::使用P=P0的人口增长公式,P0是76亿人,r是1.1%,人口达到100亿人需要多长时间?

First, substitute the given values into the formula $\begin{align*}P={P}_{0}\cdot{e}^{rt}.\end{align*}$
::首先,将给定值替换为公式P=P0ert。

$\begin{align*}10=7.6\cdot{e}^{0.011t}\end{align*}$
::10=7.6e0.011t

I solate the base and exponent by dividing both sides by 7.6.
::7.6 将双方分隔7.6,将基地隔离开来,迎头赶上。

$\begin{align*}10\:&=7.6\cdot{e}^{0.011t}\\ \small{\div}7.6 & ~~~\small{\div}7.6\\ \hline \frac{10}{7.6}&={e}^{0.011t}\end{align*}$
::10=7.6e0.011t7.67.67.6107.6=e0.011t

C ancel the base $\begin{align*}e\end{align*}$ by taking the $\begin{align*}\log_{e} \end{align*}$ or $\begin{align*}\ln\end{align*}$ of both sides of the equation.
::取消基数 e , 取出方程两侧的登录器。

$\begin{align*}\ln{\frac{10}{7.6}}&=\ln{{e}^{0.011t}}\\\\ \ln{\frac{10}{7.6}}&=0.011t\end{align*}$
::IN107.6=lne0.011tln 107.6=0.011tt

W rite the natural logarithm of a quotient as the difference of the natural logarithm of the numerator and the natural logarithm of the denominator.
::将一个商数的自然对数写为分子的自然对数和分母的自然对数的区别。

$\begin{align*}\ln{10} - \ln{7.6}&=0.011t\end{align*}$
::10-7.6=0.011吨

Finally, divide both sides by 0.011 to obtain the answer.
::最后,将双方除以0.011,以获得答案。

$\begin{align*}\ln{10} - \ln{7.6}&=0.011t\\ \div0.011 \:\:\:\:\:& \:\div0.011\\ \hline \frac{\ln{10} - \ln{7.6}}{0.011} &= t\end{align*}$
::内 - 内 - 内 - 内 - 内 - 7. 6= 0.011t 0.011 0.011 0.011n10 - 内 - 内 - 内 - 7.6. 0.011= 吨

Answer: $\begin{align*}t \approx 24.9488\end{align*}$ years
::答复:24.9488岁

E xtend all the logarithm properties to natural logs:
::将所有对数属性扩展至自然日志 :

Properties of Natural Logarithms
::自然对数属性

The Natural Logarithm of Products Property: $\begin{align*}\ln{(xy)} = \ln{x} + \ln{y}\end{align*}$
::产品属性的自然对数: In(xy) = lnx+lny

The Natural Logarithm of Quotients Property: $\begin{align*}\ln{\left(\frac{x}{y}\right)}=\ln{x} - \ln{y}\end{align*}$
::数字属性的自然对数: In(xy)=lnx-lny

The Natural Logarithm of a Power Property: $\begin{align*}\ln{x^n}=n\ln{x}\end{align*}$
::电力财产的自然对数: Inxn=nlnx

Answer the questions below to practice using logarithmic properties to solve equations.
::回答下面的问题,以便练习使用对数属性解析方程式。

Discussion Question : Use your knowledge of logarithms to prove the natural logarithm properties.
::讨论问题:使用您对数知识来证明自然对数属性。

Activity 5: Using Properties to Prove the Change of Base Formula
::活动5:利用属性证明基准公式的变化

Recall from Logarithms , that the common base for a logarithm is 10. To convert a non- common base into a common one, apply the Change-of-Base Formula.
::从 Logarithms 中回顾,对数的共同基数是 10。要将非共同基数转换成共同基数,请使用“基数变化公式”。

Formula
::公式公式公式

$\begin{align*}\log_{b}{(x)}=\frac{\log{(x)}}{\log{(b)}} \text{ or } \log_{b}{(x)}=\frac{\log_{d}{(x)}}{\log_{d}{(b)}}\end{align*}$

Now that you know the logarithm properties, the change of base formula can be explained in detail. U se the basic logarithmic equation written below in both exponential and logarithmic notation.
::既然您知道对数属性, 基公式的更改可以详细解释。使用下面以指数和对数符号书写的基本对数方程式。

$\begin{align*}x = b^y \text{ or } y = \log_{b}{x} \end{align*}$
::x=by 或 Y=logbx 键

B egin by using exponential notation.
::开始使用指数符号。

$\begin{align*}x = b^y\end{align*}$
::x=byx=byx=byxx=by x=byxx=byxxx=byxxx=byxx=byxxx=byxxx=byxxxxx=by xxxx=by byxxxx=by=by byxxx=byxxx=by byxxx=by xxxx=by by xxx=by xxx=by xxx=by=by xxx=by by xxxxx=by x=by x=by by xx=by x=by x=by xxxx=by x=by by x x=by xxx x=by by x x=by x x=by x x=by by x x x=by x x=by x x=by x x=by x x x x x=by x x x x x x x by by by x x x x x by by by x x x x x x x x x x x x x x x by by by by by x x x by by by by x x x x x x x x x x x x by by by by by by by by by by by by x x by by by by by x x x x x x by x x x x x x x x by by x x x x x x x by by by by by x by by x x x x x by by by by by by by x x x x x x by by x x x x x by by by by by x x x x by by by by by by by

Next, replace the $\begin{align*}y\end{align*}$ using the second equation, $\begin{align*}y=\log_bx.\end{align*}$
::接下来,用第二个方程y=logbx替换 Y。

$\begin{align*}x = {b}^{\left(log_{b}{x}\right)}\end{align*}$
::x=b( logbx) x=b( logbx)

Next take the log of both sides so the exponent is canceled. You will see at the end that you could use any base to do this. Let's use the common logarithm of base 10 to do this.
::下一步取出两边的日志, 这样引号就取消。您可以在结尾看到您可以使用任何基数来做到这一点。让我们使用 10 的对数来做到这一点。

$\begin{align*}\log{x}=\log{{b}^{\left(\log_{b}{x}\right)}}\end{align*}$
::logx=logb(logbx)

Using the logarithm of a power property, bring the exponent in front of the logarithm.
::使用功率属性的对数, 将推手放在对数前面。

$\begin{align*}\log{x}=\log_{b}{x}\cdot\log{b}\end{align*}$
::logx =logbx logb = logb = logb = logb = logb = logb = logb

Solving for $\begin{align*}log_{b}{x}\end{align*}$ by d ividing $\begin{align*}\log{b}\end{align*}$ to both sides will produce the change of base formula.
::通过将logb分解到两边来解决logbx, 将产生基数公式的改变。

$\begin{align*}\frac{\log{x}}{\log{b}}=\log_{b}{x}\end{align*}$

::loglogb =logbx

Use the interactive below to explore the change of base formula further.
::用下面的交互方式进一步探讨基公式的改变。

Extension: Putting It A ll T ogether
::扩展:将 " 团结在一起 " 放在一起

Use the interactive below to practice combining exponent rules.
::用下面的交互式文字来练习合并引文规则。

Wrap-Up: Review Questions
::总结:审查问题

Below is a table summarizing the properties covered in this section.
::下表概述了本节涵盖的属性。

Properties of Logs
::日志属性

Addition/Multiplication
::添加/重复

$\begin{align*}\log_bx+\log_by=\log_b(x \cdot y)\end{align*}$
::logbx+logby=logb(xy)

$\begin{align*}b^{x+y}=b^x \cdot b^y\end{align*}$
::bx+y=bxby (bx+y=bxby)

Subtraction/Division
::减/司

$\begin{align*}\log_bx+\log_by=\log_b(x \cdot y)\end{align*}$
::logbx+logby=logb(xy)

$\begin{align*}b^{x-y}=\frac{b^x}{b^y}\end{align*}$
::bx-y=bxby (x-y=bxby)

Exponentiation
::指数指数

$\begin{align*}\log_b(x^n)=n \cdot \log_bx\end{align*}$
::logb( xn) =nlogbx

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

In addition to the properties above, h ere are a few standard logarithm properties explored previously :
::除上述属性外,这里还有先前探讨的一些标准的对数属性:

$\begin{align*}\log_b 1=0\end{align*}$
::log1=0

$\begin{align*}\log_bb=1\end{align*}$
::libbb=1 logbb=1

$\begin{align*}\log_b(b^x)=x\end{align*}$
::对数b(bx)=x

$\begin{align*}b^{\log_b x}=x\end{align*}$
::博客bx=x

Use the video below to review the material in the section.

Summary
::摘要

Properties of L ogarithms:
::对数属性 :

$\begin{align*}\log_b(xy)=\log_b x +\log_b y\end{align*}$
::logb(xy) =logbx+logby(xy) =logbx+logby

$\begin{align*}log_b(x^a)=a\log_b x\end{align*}$
::logb(xa) = logbx

$\begin{align*}log_b\Big(\frac{x}{y}\Big)=\log_b x-\log_b y\end{align*}$
::logb(xy) =logbx-logby

Properties of N atural L ogarithms:
::自然对数属性 :

$\begin{align*}\ln(xy)=\ln x+\ln y\end{align*}$
::In( xy) = lnx+lny

$\begin{align*}\ln\Big(\frac{x}{y}\Big)=\ln x =\ln y\end{align*}$
::In( xy) = lnx=lny

$\begin{align*}\ln x^n = n\ln x\end{align*}$
::Inxn =nlnx

The change of base formula is $\begin{align*}\log_b(x)=\frac{log(x)}{log(b)}.\end{align*}$
::基公式的更改为logb(x)=log(x)log(b)。

章节大纲

Introduction: Seismic Activity Continued ::导言:地震活动继续

Activity 1: Logarithm of a Product Property ::活动1:产品属性的对数

The Logarithm of a Product Property ::产品属性的对数

Extension: Log Triangles ::扩展名: 日志三角

Activity 2: The Logarithm of a Power Property ::活动2:电力产权的对数

Activity 3: The Logarithm of a Quotient Property ::活动3:引数属性的对数

The Logarithm of a Quotient Property ::引用属性的对数

Activity 4: Connecting to Natural Log ::活动4:与自然日志连接

Properties of Natural Logarithms ::自然对数属性

Activity 5: Using Properties to Prove the Change of Base Formula ::活动5:利用属性证明基准公式的变化

Formula ::公式公式公式

Extension: Putting It A ll T ogether ::扩展:将 " 团结在一起 " 放在一起

Wrap-Up: Review Questions ::总结:审查问题

Properties of Logs ::日志属性

Summary ::摘要

Introduction: Seismic Activity Continued
::导言:地震活动继续

Activity 1: Logarithm of a Product Property
::活动1:产品属性的对数

The Logarithm of a Product Property
::产品属性的对数

Extension: Log Triangles
::扩展名: 日志三角

Activity 2: The Logarithm of a Power Property
::活动2:电力产权的对数

Activity 3: The Logarithm of a Quotient Property
::活动3:引数属性的对数

The Logarithm of a Quotient Property
::引用属性的对数

Activity 4: Connecting to Natural Log
::活动4:与自然日志连接

Properties of Natural Logarithms
::自然对数属性

Activity 5: Using Properties to Prove the Change of Base Formula
::活动5:利用属性证明基准公式的变化

Formula
::公式公式公式

Extension: Putting It A ll T ogether
::扩展:将 " 团结在一起 " 放在一起

Wrap-Up: Review Questions
::总结:审查问题

Properties of Logs
::日志属性

Summary
::摘要