Course: 8.8 对数的产值和引号属性

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对数的值属性和引号属性
Your friend Robbie works as a server at a pizza parlor. You and two of your friends go to the restaurant and order a pizza. You ask Robbie to bring you separate checks so you can split the cost of the pizza. Instead of bringing you three checks, Robbie brings you one with the total $\begin{align*}\log_3 162 - \log_3 2\end{align*}$ . "This is how much each of you owes," he says as he drops the bill on the table. How much do each of you owe?
::你的朋友Robbie在一家比萨饼店当服务器。你和两个朋友去餐厅订购比萨。你让Robbie给你单独支票,这样你就可以分担比萨饼的费用。Robbie给你一张支票,而不是给你三份支票,而给你一张,总账数是3162-log3/2。“这是你们每人欠的多少钱,”他说,他把帐单扔在桌子上。你们每人欠多少钱?

Product and Quotient Properties of Logarithms
::对数的值属性和引号属性

Just like exponents, logarithms have special properties, or shortcuts, that can be applied when simplifying expressions. In this lesson, we will address two of these properties.
::就像引言者一样,对数有特殊的属性或快捷键,可以在简化表达式时使用。在此课中,我们将处理其中两个属性。

Let's simplify $\begin{align*}\log_b x + \log_b y\end{align*}$ .
::让我们简化对数bx+logby。

First, notice that these logs have the same base. If they do not, then the properties do not apply.
::首先, 请注意这些日志有相同的基数。如果它们没有, 那么属性将不适用。

$\begin{align*}\log_b x=m\end{align*}$ and $\begin{align*}\log_b y=n\end{align*}$ , then $\begin{align*}b^m=x\end{align*}$ and $\begin{align*}b^n=y\end{align*}$ .
::对数bx=m和对数by=n,然后bm=x和bn=y。

Now, multiply the latter two equations together.
::现在,将后两个方程乘在一起。

$\begin{aligned} b^{m} \cdot b^{n} & = x y \\ b^{m + n} & = x y \end{aligned}$
$\begin{align*}b^m \cdot b^n &= xy \\ b^{m+n} &= xy\end{align*}$
::bmbn=xybm+n=xy

Recall, that when two exponents with the same base are multiplied, we can add the exponents. Now, reapply the logarithm to this equation.
::回顾当两个基数相同的指数乘以时,我们可以添加指数。现在,将对数重新应用到这个方程中。

$\begin{align*}b^{m+n}=xy \rightarrow \log_b xy=m+n\end{align*}$
::bm+n=xy=xlogb=xy=m+n=m+n=xy=m+n=xy=xlogb=xy=m+n=m+n=m=m+n=xy=m+n=xy=m+n=xy=xy=xy@xy=xy=m@xy=m@xy=m+n

Recall that $\begin{align*}m=\log_b x\end{align*}$ and $\begin{align*}n=\log_b y\end{align*}$ , therefore $\begin{align*}\log_b xy=\log_b x + \log_b y\end{align*}$ .
::回顾 m=logbx 和 n=logby, 因此logbxy=logbx+logby 。

This is the Product Property of Logarithms .
::这是对数的产品属性。

Now, let's expand $\begin{align*}\log_{12} 4y\end{align*}$ .
::现在,让我们扩展日志 124y。

Applying the Product Property from the previous problem , we have:
::在应用上一个问题的产品属性时,我们有:

$\begin{align*}\log_{12} 4y = \log_{12} 4 + \log_{12} y\end{align*}$
::对数 12\\\4y=log12}4+log12y

Finally, let's simplify $\begin{align*}\log_3 15 - \log_3 5\end{align*}$ .
::最后,让我们简化对数3+15-log3+5。

As you might expect, the Quotient Property of Logarithms is $\begin{align*}\log_b \frac{x}{y}=\log_b x - \log_b y\end{align*}$ (proof in the Review section ). Therefore, the answer is:
::正如你可能预计的那样,对数的引号属性为logbxy=logbx-logby(在审查部分中为校验)。

$\begin{aligned} \log_{3} 15 - \log_{3} 5 & = \log_{3} \frac{15}{5} \\ = \log_{3} 3 \\ = 1 \end{aligned}$
$\begin{align*}\log_3 15 - \log_3 5 &= \log_3 \frac{15}{5} \\ &= \log_3 3 \\ &= 1\end{align*}$
::对数 3\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the amount that each of you owes.
::早些时候,有人要求你找出你们每人欠的金额

If you rewrite $\begin{align*}\log_3 162 - \log_3 2\end{align*}$ as $\begin{align*}\log_3 \frac {162}{2}\end{align*}$ , you get $\begin{align*}\log_3 81\end{align*}$ .
::如果您重写日志3162- log32为日志31622, 您将得到日志381 。

$\begin{align*}3^4 = 81\end{align*}$ so you each owe $4.
::34=81 所以你们每人欠4美元

Example 2
::例2

Simplify the following expression: $\begin{align*}\log_7 8 + \log_7 x^2 + \log_7 3y\end{align*}$ .
::简化以下表达式 : log7+8+log7+x2+log7+3y 。

Combine all the logs together using the Product Property.
::使用产品属性将所有日志合并在一起。

$\begin{aligned} \log_{7} 8 + \log_{7} x^{2} + \log_{7} 3 y & = \log_{7} 8 x^{2} 3 y \\ = \log_{7} 24 x^{2} y \end{aligned}$
$\begin{align*}\log_7 8 + \log_7 x^2 + \log_7 3y &= \log_7 8x^2 3y \\ &= \log_7 24x^2 y\end{align*}$
::对数 7\\\ 8+log7\\ x2+log7\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Example 3
::例3

Simplify the following expression: $\begin{align*}\log y - \log 20 + \log 8x\end{align*}$ .
::简化以下表达式: logy-log20+log8x。

Use both the Product and Quotient Property to condense.
::使用产品和数字属性来压缩。

$\begin{aligned} \log y - \log 20 + \log 8 x & = \log \frac{y}{20} \cdot 8 x \\ = \log \frac{2 x y}{5} \end{aligned}$
$\begin{align*}\log y - \log 20 + \log 8x &= \log \frac{y}{20} \cdot 8x \\ &= \log \frac{2xy}{5}\end{align*}$
::logy_log_log_20+log_8x=logy20_8x=log_2xy5

Example 4
::例4

Simplify the following expression: $\begin{align*}\log_2 32 - \log_2 z\end{align*}$ .
::简化以下表达式: log232- log2z。

Be careful; you do not have to use either rule here, just the definition of a logarithm.
::注意; 这里不需要使用任何规则, 只需使用对数定义。

$\begin{aligned} \log_{2} 32 - \log_{2} z = 5 - \log_{2} z \end{aligned}$
$\begin{align*}\log_2 32 - \log_2 z=5 - \log_2 z\end{align*}$
::log2\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Example 5
::例5

Simplify the following expression: $\begin{align*}\log_8 \frac{16x}{y^2}\end{align*}$ .
::简化以下表达式 : log8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

When expanding a log, do the division first and then break the numerator apart further.
::当扩展日志时, 请先执行分区, 然后再进一步拆分分子数。

$\begin{aligned} \log_{8} \frac{16 x}{y^{2}} & = \log_{8} 16 x - \log_{8} y^{2} \\ = \log_{8} 16 + \log_{8} x - \log_{8} y^{2} \\ = \frac{4}{3} + \log_{8} x - \log_{8} y^{2} \end{aligned}$
$\begin{align*}\log_8 \frac{16x}{y^2} &= \log_8 16x - \log_8 y^2 \\ &= \log_8 16 + \log_8 x-\log_8 y^2 \\ &= \frac{4}{3} + \log_8 x - \log_8 y^2\end{align*}$
::log8\\\ 16xy2 = log8\\ 16x- log8\\\ y2 = log8\ 16+ log8\\ x- log8\\ log8\\ y2 = 43+ log8\ x- log8\ y2

To determine $\begin{align*}\log_8 16\end{align*}$ , use the definition and powers of 2: $\begin{align*}8^n=16 \rightarrow 2^{3n}=2^4 \rightarrow 3n = 4 \rightarrow n=\frac{4}{3}\end{align*}$ .
::为确定log816, 使用定义和权限为 2: 8n=1623n=243n=4n=43。

Review
::回顾

Simplify the following logarithmic expressions.
::简化以下对数表达式。

$\begin{align*}\log_3 6 + \log_3 y - \log_3 4\end{align*}$
::对数 3\\\ 6+log3\ y- log3\ 4

$\begin{align*}\log 12 - \log x + \log y^2\end{align*}$
::log_log_x+logy2

$\begin{align*}\log_6 x^2 - \log_6 x - \log_6 y\end{align*}$
::对数 6\\\ x2 - log6\\ x- log6\\ y

$\begin{align*}\ln 8 + \ln 6 - \ln 12\end{align*}$
::In8+ln_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{align*}\ln 7 - \ln 14 + \ln 10\end{align*}$
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}...

$\begin{align*}\log_{11} 22 + \log_{11} 5 - \log_{11} 55\end{align*}$
::log11\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Expand the following logarithmic functions.
::展开以下对数函数。

$\begin{align*}\log_6 (5x)\end{align*}$
::对数 6\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

$\begin{align*}\log_3 (abc)\end{align*}$
::对数3(abc)

$\begin{align*}\log \left(\frac{a^2}{b}\right)\end{align*}$
::对数(a2b)

$\begin{align*}\log_9 \left(\frac{xy}{5}\right)\end{align*}$
::对数 9(xy5)

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

$\begin{align*}\log \left(\frac{8x^2}{15}\right)\end{align*}$
::log (8x215)

$\begin{align*}\log_4 \left(\frac{5}{9y}\right)\end{align*}$
::对数 4( 59y)

Write an algebraic proof of the Quotient Property. Start with the expression $\begin{align*}\log_a x - \log_a y\end{align*}$ and the equations $\begin{align*}\log_a x=m\end{align*}$ and $\begin{align*}\log_a y=n\end{align*}$ in your proof. Refer to the proof of the Product Property in the first practice problem as a guide for your proof.
::写入引用属性的代数证明。以表达式logax- logay 和公式logax=m 和logay=n 开始。请参考第一个练习问题中的产品属性证明作为证明指南。

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

对数的值属性和引号属性

Product and Quotient Properties of Logarithms ::对数的值属性和引号属性

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)

Product and Quotient Properties of Logarithms
::对数的值属性和引号属性

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)