Course: 8.5 定义对数 | The Way To Learn

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定义对数
You go a concert and you want to know how loud it is in decibels. The decibel level of a sound is found by first assigning an intensity I0 to a very soft sound, or the threshold. The decibel level can then be measured with the formula $\begin{align*}d = 10 \cdot \log \frac{I}{I0}\end{align*}$ where I is the intensity of the sound. If the intensity of the concert is 1,000,000,000(I0), what is its decibel level?
::您去听音乐会, 您想知道音乐会在分贝中有多响。音响的分贝级别是通过先将强度 I0 指定为非常柔软的音频或阈值来找到的。然后, 分贝级别可以用公式 d= 10 log II0 来测量, 在那里我就是声音的强度。如果音乐会的强度是1000 000,000,000 (I0) , 那么它的分贝级别是多少 ?

Logarithm
::对数对数

You can probably guess that $\begin{align*}x=3\end{align*}$ in $\begin{align*}2^x=8\end{align*}$ and $\begin{align*}x=4\end{align*}$ in $\begin{align*}2^x=16\end{align*}$ . But, what is $\begin{align*}x\end{align*}$ if $\begin{align*}2^x=12\end{align*}$ ? Until now, we did not have an inverse to an exponential function. But, because we have variables in the exponent, we need a way to get them out of the exponent. We will now introduce the logarithm. A logarithm is defined as the inverse of an exponential function. It is written $\begin{align*}\log_b a=x\end{align*}$ such that $\begin{align*}b^x=a\end{align*}$ . Therefore, if $\begin{align*}5^2=25\end{align*}$ ( exponential form ), then $\begin{align*}\log_5 25=2\end{align*}$ ( logarithmic form ).
::您可以猜测 2x=8 中的 x= 3 和 2x= 16 中的 x= 4 。但是, 如果 2x= 12, 那么, 是什么 x ? 直到现在, 我们没有反向指数函数。但是, 因为我们在指数中有变量, 我们需要一种方法来把它们从指数中调出。我们现在将引入对数。对数被定义为指数函数的反向值。这是写入的对数bx=a=x, bx=a 。因此, 如果是52=25( Excential form) , 然后是log5\\ 25=2( logatrict form) 。

There are two special logarithms, or logs. One has base 10, and rather that writing $\begin{align*}\log_{10}\end{align*}$ , we just write log. The other is the natural log , the inverse of the natural number. The natural log has base $\begin{align*}e\end{align*}$ and is written $\begin{align*}\ln\end{align*}$ . This is the only log that is not written using $\begin{align*}\log\end{align*}$ .
::有两种特殊的对数或日志。一种是基数 10, 另一种是写日志 10, 我们只需写日志。另一种是自然日志, 自然日志的反数。自然日志的基数 e, 并用日志写入。这是唯一没有使用日志写入的日志。

Let's rewrite $\begin{align*}\log_3 27=3\end{align*}$ in exponential form.
::让我们重写对数 3\\\\ 27= 3 的指数形式。

Use the definition above, also called the “key”.
::使用上述定义,也称为“钥匙”。

$\begin{aligned} \log_{b} a & = x \leftrightarrow b^{x} = a \\ \log_{3} 27 & = 3 \leftrightarrow 3^{3} = 27 \end{aligned}$
$\begin{align*}\log_b a &= x \leftrightarrow b^x=a \\ \log_3 27 &= 3 \leftrightarrow 3^3=27 \end{align*}$
::logba=xbx=alog3=27=333=27

Now, let's find the following.
::现在,让我们来找找。

$\begin{align*}\log 1000\end{align*}$
::对数1000

$\begin{align*}\log 1000=x \Rightarrow 10^x=1000, x=3\end{align*}$ .
::对数=1 000=x=10x=1000x=3

$\begin{align*}\log_7 \frac{1}{49}\end{align*}$
::对数7149

$\begin{align*}\log_7 \frac{1}{49}=x \Rightarrow 7^x=\frac{1}{49}, x=-2\end{align*}$ .
::log7149=x7x=149,x2。

$\begin{align*}\log_{\frac{1}{2}}(-8)\end{align*}$
::对数 12(- 8)

$\begin{align*}\log_{\frac{1}{2}}(-8)=x \Rightarrow \left(\frac{1}{2}\right)^x =-8\end{align*}$ . There is no solution. A positive number when raised to any power will never be negative.
::log12(- 8) =x( 12) x8. 不存在解决方案。向任何权力提升的正数永远不会是负数。

Using the key, we can rearrange all of these in terms of exponents.
::使用钥匙,我们可以重新排列所有的这些在引言。

There are two special logarithms that you may encounter while writing them into exponential form.
::在将它们写成指数形式时,可能会遇到两种特殊的对数。

The first is $\begin{align*}\log_b 1=0\end{align*}$ , because $\begin{align*}b^0=1\end{align*}$ . The second is $\begin{align*}\log_b b=1\end{align*}$ because $\begin{align*}b^1=b \cdot b\end{align*}$ can be any number except 1.
::第一个是logb1=0,因为 b0=1.第二个是logbb=1,因为 b1=bb可以是除1以外的任何数字。

Finally, let's use a calculator to find the following logarithms and round our answers to the nearest hundredth.
::最后,让我们用计算器找到以下的对数并绕过我们的答案到最近的一百个。

$\begin{align*}\ln7\end{align*}$
::内 7

Locate the LN button on your calculator. Depending on the brand, you may have to input the number first. For a TI-83 or 84, press LN , followed by the 7 and ENTER . The answer is 1.95.
::在您的计算器上找到 LN 按钮。根据品牌, 您可能需要先输入数字。对于 TI- 83 或 84, 请按 LN 键, 然后是 7 和 ENTER 。答案是 1. 95 。

$\begin{align*}\log 35\end{align*}$
::对数35

The LOG button on the calculator is base 10. Press LOG , 35, ENTER . The answer is 1.54.
::计算器上的LOG按钮是10号基地,请按LOG,35号,ENTER,答案是1.54

$\begin{align*}\log_5 226\end{align*}$
::对数5226

To use the calculator for a base other than 10 or the natural log, you need to use the formula.
::要使用除 10 或自然日志以外的基数计算器,您需要使用公式。

Change of Base Formula: $\begin{align*}\log_a x=\frac{\log_b x}{\log_b a}\end{align*}$ , such that $\begin{align*}x, a,\end{align*}$ and $\begin{align*}b>0\end{align*}$ and $\begin{align*}a\end{align*}$ and $\begin{align*}b \ne 1\end{align*}$ .
::基数公式变化:logax=logbxlogba, 如 x,a, b>0, a 和 b1 。

So, to use this for a calculator, you can use either LN or LOG.
::所以,要用这个来计算计算, 您可以使用 LN 或 LOG 。

$\begin{align*}\log_5 226=\frac{\log 226}{\log 5}\end{align*}$ or $\begin{align*}\frac{\ln 226}{\ln 5} \approx 3.37\end{align*}$
::=================================================================================================================================================================5===========================================================================================================================================================================================================================================================================================================================================================

In the TI-83 or 84, the keystrokes would be LOG (226)/ LOG (5), ENTER .
::在TI-83或84中,键盘为LOG(226)/LOG(5),ENTER。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the decibel level of a concert if the intensity is 1,000,000,000(I0).
::早些时候,有人要求你找到音乐会的分贝级别,如果强度是1000 000 000(I0)的话。

Plug the given values into the equation $\begin{align*}d = 10 \cdot \log \frac{I}{I0}\end{align*}$ and solve for d .
::将给定值插入方程式 d=10 log II0 并解析 d 。

$\begin{aligned} d = 10 \cdot \log \frac{1, 000, 000, 000 (I 0)}{I 0} \\ d = 10 \cdot \log 1, 000, 000, 000 \\ d = 10 \cdot 9 = 90 \end{aligned}$
$\begin{align*}d = 10 \cdot \log \frac{1,000,000,000 (I0)}{I0}\\ d = 10 \cdot \log 1,000,000,000\\ d = 10 \cdot 9 = 90\end{align*}$
::d=10 10 log=1000 000 000 000(I0)I0d=10 log=1000 000 000d=109=90

Therefore, the decibel level of the concert is 90.
::因此,音乐会的分贝级别是90。

Example 2
::例2

Write $\begin{align*}6^2=36\end{align*}$ in logarithmic form.
::以对数形式写入 62=36 。

Using the key, we have: $\begin{align*}6^2=36 \rightarrow \log_6 36=2\end{align*}$ .
::使用键,我们有: 62=36 log636=2。

For Examples 3-5, evaluate the expressions without a calculator.
::实例3-5,评价不使用计算器的表达式。

Change each logarithm into exponential form and solve for $\begin{align*}x\end{align*}$ .
::将每个对数修改为指数形式并解决 x 。

Example 3
::例3

$\begin{align*}\log_{\frac{1}{2}} 16\end{align*}$
::log1216

$\begin{align*}\log_{\frac{1}{2}} 16 \rightarrow \left(\frac{1}{2}\right)^x=16\end{align*}$ . $\begin{align*}x\end{align*}$ must be negative because the answer is not a fraction, like the base.
::log1216(12)x=16. x 必须为负值, 因为答案不是一个分数, 像基数一样。

$\begin{align*}2^4=16\end{align*}$ , so $\begin{align*}\left(\frac{1}{2}\right)^{-4}=16\end{align*}$ . Therefore, $\begin{align*}\log_{\frac{1}{2}}16=-4\end{align*}$ .
::24=16, 所以(12) -4=16, 因此, log12+16+4。

Example 4
::例4

$\begin{align*}\log 100\end{align*}$
::对数100

$\begin{align*}\log100 \rightarrow 10^x=100\end{align*}$ . $\begin{align*}x=2\end{align*}$ , therefore, $\begin{align*}\log100=2\end{align*}$ .
::log 10010x=100. x=2,因此,log 100=2。

Example 5
::例5

$\begin{align*}\log_{64} \frac{1}{8}\end{align*}$
::对数6418

$\begin{align*}\log_{64} \frac{1}{8} \rightarrow 64^x = \frac{1}{8}\end{align*}$ . First, $\begin{align*}\sqrt{64}=8\end{align*}$ , so $\begin{align*}64^{\frac{1}{2}}=8\end{align*}$ . To make this a fraction, we need to make the power negative. $\begin{align*}64^{-\frac{1}{2}}=\frac{1}{8}\end{align*}$ , therefore $\begin{align*}\log_{64} \frac{1}{8}=-\frac{1}{2}\end{align*}$ .
::log641864x=18。首先, 64=8, 所以6412=8。要将这个分数变成一个分数, 我们需要将功率为负。 64- 12=18, 因此是log641812 。

Example 6
::例6

Use the change of base formula to evaluate $\begin{align*}\log_8 \frac{7}{9}\end{align*}$ in a calculator.
::使用基公式的修改来评估计算器中的对log879。

Rewriting $\begin{align*}\log_8 \frac{7}{9}\end{align*}$ using the change of base formula, we have: $\begin{align*}\frac{\log \frac{7}{9}}{\log 8}\end{align*}$ . Plugging it into a calculator, we get $\begin{align*}\frac{\log \left(\frac{7}{9}\right)}{\log 8} \approx -0.12\end{align*}$ .
::使用基公式的更改重写对数879,我们有:对数79log8。将其插入计算器,我们得到对数*(79)log80.12。

Review
::回顾

Convert the following exponential equations to logarithmic equations.
::将以下指数方程式转换为对数方程式。

$\begin{align*}3^x=5\end{align*}$
::3x=5

$\begin{align*}a^x=b\end{align*}$
::ax=b

$\begin{align*}4(5^x)=10\end{align*}$
::4(5x)=10

Convert the following logarithmic equations to exponential equations.
::将以下对数方程式转换为指数方程式。

$\begin{align*}\log_2 32=x\end{align*}$
::log2\32=x

$\begin{align*}\log_{\frac{1}{3}}x=-2\end{align*}$
::对数 13x2

$\begin{align*}\log_a y=b\end{align*}$
::logay=b

Convert the following logarithmic expressions without a calculator.
::将下列对数表达式转换为无计算器的对数表达式。

$\begin{align*}\log_5 25\end{align*}$
::对数 5 @% 25

$\begin{align*}\log_{\frac{1}{3}} 27\end{align*}$
::log13\\\27

$\begin{align*}\log \frac{1}{10}\end{align*}$
::log 110

$\begin{align*}\log_2 64\end{align*}$
::log264

Evaluate the following logarithmic expressions using a calculator. You may need to use the Change of Base Formula for some problems.
::使用计算器对以下对数表达式进行评估。您可能需要使用“ 基数修改公式” 来应对一些问题。

$\begin{align*}\log 72\end{align*}$
::对数72

$\begin{align*}\ln 8\end{align*}$
::内 8

$\begin{align*}\log_2 12\end{align*}$
::log2\12

$\begin{align*}\log_3 9\end{align*}$
::对数 3 @% 9

$\begin{align*}\log_{11} 32\end{align*}$
::log1132

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

定义对数

Logarithm ::对数对数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

log ⁡ 100 \begin{align*}\log 100\end{align*} ::对数100

Example 5 ::例5

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)

Logarithm
::对数对数

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

$\begin{align}\log 100\end{align}$
::对数100

Example 5
::例5

Example 6
::例6

Review
::回顾

Review (Answers)
::回顾(答复)