Course: 11.6 评价对数 | The Way To Learn

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评估对数
The NCAA basketball tournament has 64 teams (after they play in games). The games are between two teams at a time. Using this information, we can find the number of rounds necessary to determine a winner of the tournament ¹ .
::NCAA篮球锦标赛有64个球队(在比赛后),比赛由两个球队同时进行。利用这些信息,我们可以找到确定比赛获胜者所需的回合数。

To do this we will use a logarithm, and we cover how to evaluate those in this section.
::为此,我们将使用对数,并涵盖如何评估本节中的内容。

What Is a Logarithm?

::什么是对数?

For the previous functions we have covered, it has been clear what the inverse operation was: addition and subtraction, multiplying and dividing, squaring and the square root. However, now that the variable is in the exponent, it is not clear what the inverse operation is. We need to define one, so we introduce the logarithm. A logarithm is defined as the inverse of an exponential statement.
::对于我们所覆盖的先前函数, 已经很清楚反向操作是: 加和减、乘和分、平方根。但是, 既然变量在指数中, 则不清楚反向操作是什么。我们需要定义一个, 所以我们要引入对数。对数被定义为指数式语句的反向。

Logarithmic Functions
::对对数函数

The logarithmic function is $\begin{aligned} y = \log_{b} x, \end{aligned}$ and is equivalent to $\begin{aligned} b^{y} = x, \end{aligned}$ where $\begin{aligned} x > 0 \end{aligned}$ and $\begin{aligned} b > 0, b \neq 1 \end{aligned}$ .

The logarithm above is said "log-base- b of x equals y ." The value of the subscript b is the base of the logarithm, just as b is the base in the exponential expression $\begin{aligned} b^{x} \end{aligned}$ .
::上面的对数是“ x 的 log- b 基数等于 y ” 。下标 b 的值是对数的基数, 正如 b 是指数表达式 bx 中的基数一样。

Converting Between Exponential Form and Logarithmic Form
::对数表和对数表之间的转换

Since the logarithm is the inverse function, we should be able to switch the values of $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ to get a logarithmic statement from an exponential statement. Essentially, that is what we do. We identify the base, the exponent, and the result (of raising a base to a power; note this is the argument of the logarithm) and arrange them in a logarithmic statement.
::由于对数是反函数, 我们应该能够切换 x 和 y 的值, 从指数语句中获取对数语句。基本上, 这就是我们要做的。我们确定基点、指数和结果( 将基点提升到功率; 注意这是对数的参数) , 并将它们安排在对数语句中。

For example, if $\begin{aligned} 5^{2} = 25 \end{aligned}$ , a statement in exponential form , the base is 5, the exponent is 2, and the result is 25. The statement in logarithmic form is $\begin{aligned} \log_{5} 25 = 2 \end{aligned}$ . Notice for the logarithm we input the results (that is, the results are arguments) and we get out exponents.
::例如,如果52=25, 一个指数式的语句, 基数为5, 引号为2, 结果为25, 对数式的语句是log525=2, 我们输入结果的对数通知( 即结果为参数) , 然后我们出出引号。

Example 1
::例1

Write the following exponential statements in logarithmic form:
::以对数形式写入下列指数语句:

a. $\begin{aligned} 6^{2} = 36 \end{aligned}$
::a. 62=36

b. $\begin{aligned} (\frac{1}{4})^{- 3} = 64 \end{aligned}$
::b. (14-3=64)

c. $\begin{aligned} p^{r} = s \end{aligned}$
::c. pr=s

Solution:
::解决方案 :

a. We need to identify the base, 6, the exponent, 2, and the result, 36. We have $\begin{aligned} 6^{2} = 36 \leftrightarrow \log_{6} 36 = 2 \end{aligned}$ .
::a. 我们需要确定基数6,前言,2,结果36,我们有62=36, log6,36=2。

b. The base is $\begin{aligned} \frac{1}{4} \end{aligned}$ , the exponent is $\begin{aligned} - 3 \end{aligned}$ , and the result is 64. $\begin{aligned} (\frac{1}{4})^{- 3} = 64 \leftrightarrow \log_{\frac{1}{4}} 64 = - 3 \end{aligned}$
::b. 基数为14, 指数为 -3, 结果为64 (14-3=64 log1464=3)

c. The base is $\begin{aligned} p \end{aligned}$ , the exponent is $\begin{aligned} r \end{aligned}$ , and the result is $\begin{aligned} s \end{aligned}$ . $\begin{aligned} p^{r} = s \leftrightarrow \log_{p} s = r \end{aligned}$
::c. 基数为 p, 出处为 r, 结果为 s. pr=s logps=r

Example 2
::例2

Write the following logarithmic statements in exponential form:
::以指数形式写入下列对数语句:

a. $\begin{aligned} \log_{3} 27 = 3 \end{aligned}$
::a. 日数3=27=3

b. $\begin{aligned} \log_{49} 7 = \frac{1}{2} \end{aligned}$
::b. 日数49=7=12

c. $\begin{aligned} \log_{j} k = l \end{aligned}$
::c. logjk=l

Solution:
::解决方案 :

a. As above we need to identify the base, the exponent, and the result. The base is 3, the exponent is also 3, and the result is 27. $\begin{aligned} \log_{3} 27 & = 3 \leftrightarrow 3^{3} = 27 \end{aligned}$
::a. 如上所示,我们需要确定基数、指数和结果。基数为3,指数也是3,结果为27,log3______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

b. The base is 49, the exponent is $\begin{aligned} \frac{1}{2} \end{aligned}$ , and the result is 7. $\begin{aligned} \log_{49} 7 = \frac{1}{2} \leftrightarrow 49^{\frac{1}{2}} = 7 49^{\frac{1}{2}} = \sqrt{49} = 7 \end{aligned}$
::b. 基数为49,指数为12,结果为7.log49=7=12=4912=7 4912=4912=49=7

c. The base is $\begin{aligned} j \end{aligned}$ , the exponent is $\begin{aligned} l \end{aligned}$ , and the result is $\begin{aligned} k \end{aligned}$ . $\begin{aligned} \log_{j} k = l \leftrightarrow j^{l} = k \end{aligned}$
::c. 基数为j, 出处为l, 结果为 k. logjk=ljl=k

by HCCMathHelp shows several examples of converting between exponential and logarithmic forms.
::HCCMathHelp展示了几例指数形式和对数形式之间的转换。

Evaluating Logarithms
::评估对数

When you evaluate logarithms, it is helpful to think of them in exponential form. Remember that for logarithms, we input results and get out exponents. Logarithms are equal to numbers.
::当您评估对数时, 以指数形式来看待它们是有益的。记住, 对于对数, 我们输入结果, 并让出指数。对数等于数字。

(Author's Note: I will avoid using a variable in place of the exponent when we think of logarithms in exponential form. This should help avoid confusion regarding "solving" logarithms and the idea that logarithms are equations—instead of numbers—as you often see in other texts.)
:作者注:当我们想到指数形式的对数时,我将避免使用变量代替指数。这应有助于避免在“解决”对数和对数是方程——而不是数字的观念上产生混淆,正如你在其他文本中经常看到的那样。 )

Example 3
::例3

Evaluate the following logarithms:
::评估下列对数:

a. $\begin{aligned} \log_{7} \frac{1}{49} \end{aligned}$
::a. 日对数7+149

b. $\begin{aligned} \log_{\frac{1}{2}} (- 8) \end{aligned}$
::b. 对数12(-8)

c. $\begin{aligned} \log_{\frac{1}{2}} 16 \end{aligned}$
::c. 日数1216

d. $\begin{aligned} \log_{64} \frac{1}{8} \end{aligned}$
::d. 日数6418

Solution: It will be easier to think of these logarithms in exponential form.
::解答: 比较容易以指数形式想到这些对数。

a. $\begin{aligned} \log_{7} \frac{1}{49} ⟹ 7^{?} = \frac{1}{49} \end{aligned}$ . The question here is, what can we raise 7 to to get $\begin{aligned} \frac{1}{49} \end{aligned}$ back? Note that 49 is 7 squared and it is in the denominator, so $\begin{aligned} \log_{7} \frac{1}{49} = - 2 \end{aligned}$ .
::a. log71497=7?=149。这里的问题是,我们怎样才能筹集到7来得到149?请注意,49是7平方,在分母中,所以log7149=2。

b. $\begin{aligned} \log_{\frac{1}{2}} (- 8) ⟹ {(\frac{1}{2})}^{?} = - 8 \end{aligned}$ . This is impossible. A positive number raised to any power will never be negative. See the warning box below.
::b. log12_( 8)_( 12)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\D\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\这是不可能的。这是不可能的。这不可能。向任何权力提出的正数不会是负数。。。任何权力不会是负数。见下面的警告框。

c. $\begin{aligned} \log_{\frac{1}{2}} 16 ⟹ {(\frac{1}{2})}^{?} = 16 \end{aligned}$ . The power must be negative because the result is not a fraction . If we focus on 2, $\begin{aligned} 2^{4} = 16 \end{aligned}$ , so $\begin{aligned} {(\frac{1}{2})}^{- 4} = 16 \end{aligned}$ . Therefore, $\begin{aligned} \log_{\frac{1}{2}} 16 = - 4 \end{aligned}$ .
::c. log12161616?=16. 功率必须是负的, 因为结果不是一个分数。如果我们聚焦于 2, 24=16, 所以(12)-4=16。因此, log1216=4。

d. $\begin{aligned} \log_{64} \frac{1}{8} ⟹ 64^{?} = \frac{1}{8} \end{aligned}$ . First, $\begin{aligned} \sqrt{64} = 8 \end{aligned}$ , so $\begin{aligned} 64^{\frac{1}{2}} = 8 \end{aligned}$ . To get the 8 in the denominator , we need to make the power negative. $\begin{aligned} 64^{- \frac{1}{2}} = \frac{1}{8} \end{aligned}$ , therefore, $\begin{aligned} \log_{64} \frac{1}{8} = - \frac{1}{2} \end{aligned}$ .
::d. log641864?=18。首先, 64=8, 所以6412=8。要获得分母中的 8, 我们需要将电量为负。 64-12=18, 因此, log6418=12。

WARNING
::警告

You cannot take the logarithm of a negative number or zero. For example, $\begin{aligned} 2^{- 3} \neq - 8 \end{aligned}$ .

by CK-12 gives an example of evaluating a logarithm.
::在 CK-12 中给出了对数评估的示例。



Example 4
::例4

The NCAA basketball tournament has 64 teams (after they play in games). The games are between two teams at a time ¹ . Using this information, find the number of rounds necessary to determine a winner of the tournament.
::NCAA篮球锦标赛有64个球队(在比赛后),每场比赛由两个球队一组进行,1 利用这一信息,找到确定比赛优胜者所需的回合数。

Solution: 64 is the result of pairs of teams playing a certain number of rounds. We can find the logarithm base 2 of 64 to determine the number of rounds.
::解答: 64 是两组球队玩数轮的结果。我们可以找到64的对数基 2 来决定弹数。

$\begin{aligned} \log_{2} 64 ⟹ 2^{?} = 64 \\ \log_{2} 64 = 6 \end{aligned}$
There are 6 rounds of the NCAA basketball tournament.
::=64log264=6 有6轮NCAA篮球锦标赛

by Math & Science 2024 shows several examples of evaluating logarithms. (Note: The idea regarding fractions yielding negative exponents works only if the base is a natural number.)
:注:关于产生负指数的分数的想法只有在基数为自然数字时才有效。 )



Special Logarithms
::特殊对数

There are two special logarithms, or logs, for short. One is the log base 10, which we call the common logarithm because it is commonly used in our base 10 number system. Rather than writing $\begin{aligned} \log_{10} M \end{aligned}$ , we just write $\begin{aligned} \log M \end{aligned}$ . The other special log is the natural log arithm , which is log base $\begin{aligned} e \end{aligned}$ and is written $\begin{aligned} \ln \end{aligned}$ . This is the only log that is not written using $\begin{aligned} \log \end{aligned}$ .
::简称有两个特殊的对数或日志。其中之一是对数基准 10, 我们称之为对数基准 10, 因为它在我们的对数基准 10 系统中常用。我们没有写入对数 10\\\ M, 而是写入对数 \\ M。另一种特殊日志是自然对数, 即对数基准 e e , 并写入。这是唯一没有使用日志写入的对数。

Common Logarithm and Natural Logarithm
::普通对数和自然对数

$\begin{aligned} Common Logarithm & Natural Logarithm \\ \log_{10} x = \log x & \log_{e} x = \ln x \end{aligned}$

Example 5
::例5

Evaluate the following:
::评估以下方面:

a. $\begin{aligned} \log 1, 000 \end{aligned}$
::a. log%1 000

b. $\begin{aligned} \log 0.01 \end{aligned}$
::b. log0.01

Solution:
::解决方案 :

a. $\begin{aligned} \log 1, 000 ⟹ 10^{?} = 1, 000, \log 1, 000 = 3 \end{aligned}$ .
::a. log1 00010? = 1 000,log1 000=3。

b. $\begin{aligned} \log 0.01 ⟹ 10^{?} = 0.01 = \frac{1}{100}, \log 0.01 = - 2 \end{aligned}$ .
::b. log0.0110? =0.01=1100. log0.012。

by Mathispower4u shows how to evaluate common logarithms without a calculator.
::Mathispower4u 展示了如何在没有计算器的情况下评估通用对数。

Example 6
::例6

Evaluate the following:
::评估以下方面:

a. $\begin{aligned} \ln e^{4} \end{aligned}$
::a. 内额 4

b. $\begin{aligned} \ln \sqrt[3]{e} \end{aligned}$
::b. 内额 3

Solution:
::解决方案 :

a. $\begin{aligned} \ln e^{4} ⟹ e^{?} = e^{4}, \ln e^{4} = 4 \end{aligned}$
::a. e4 e? = e4, e4=4 e4 e4=4

b. $\begin{aligned} \ln \sqrt[3]{e} ⟹ e^{?} = \sqrt[3]{e} = e^{\frac{1}{3}}, \ln \sqrt[3]{e} = \frac{1}{3} \end{aligned}$
::=e3=e13 =13 =13 =13 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Change of Base Formula
::基基公式变动

The following formula can be helpful especially when trying to evaluate logarithms with a calculator:
::下列公式尤其有助于用计算器评估对数:

Formula
::公式公式公式

$\begin{aligned} \log_{a} x = \frac{\log_{b} x}{\log_{b} a}, \end{aligned}$
where $\begin{aligned} x, a, \end{aligned}$ and $\begin{aligned} b > 0 \end{aligned}$ and $\begin{aligned} a, b \neq 1 \end{aligned}$ .
::loga_x=logb_xlogb_a, x,a,和b>0和a,b_1。

Example 7
::例7

Use your calculator to find the logarithms below. Round your answer to the nearest hundredth.
::使用您的计算器查找下面的对数。将您的答复四舍五入到最接近的百位。

a. $\begin{aligned} \ln 7 \end{aligned}$
::a. 内 7

b. $\begin{aligned} \log 35 \end{aligned}$
::b. log_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

c. $\begin{aligned} \log_{5} 226 \end{aligned}$
::c. 日数5226

d . $\begin{aligned} \log_{7} 94 \end{aligned}$
::d. 日数794

e. $\begin{aligned} \log_{8} \frac{7}{9} \end{aligned}$
::e. log8_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Solution:
::解决方案 :

a. Locate the LN button on your calculator. Depending on the brand, you may have to input the number 1st. $\begin{aligned} \ln 7 = 1.95 \end{aligned}$
::a. 在您的计算器上定位 LN 按钮。取决于品牌, 您可能需要输入数字 1。 In7=1. 95

b. The LOG button on the calculator is base 10. $\begin{aligned} \log 35 = 1.54 \end{aligned}$
::b. 计算器上的LOG按钮为基数 10,log=35=1.54。

c. To use the calculator for a base other than 10 or the natural log, you need to use the change of base formula. To use this for a calculator, you can use either LN or LOG (or any base).
::c. 要使用除10个基数或自然日志以外的基数的计算器,您需要使用基数公式的改变。要使用这个计算器,您可以使用LN或LOG(或任何基数)。

$\begin{aligned} \log_{5} 226 & = \frac{\log 226}{\log 5} = 3.37 \\ or & = \frac{\ln 226}{\ln 5} = 3.37 \end{aligned}$
d. Since we can use either log base 10 or log base $\begin{aligned} e \end{aligned}$ , we will use $\begin{aligned} \ln \end{aligned}$ .
::log5226=log226log@5=3.37or=ln226ln_5=3.37d。由于我们可以使用对数基数 10 或对数基数 e,我们将使用 In。

$\begin{aligned} \log_{7} 94 = \frac{\ln 94}{\ln 7} = 2.33 \end{aligned}$

::对数794=ln94ln7=2.33

e. Rewriting $\begin{aligned} \log_{8} \frac{7}{9} \end{aligned}$ using the change of base formula with base 10, we have
::e. 使用以基数10为基数的基数公式变化来重写对log8 79,我们

$\begin{aligned} \log_{8} \frac{7}{9} = \frac{\log \frac{7}{9}}{\log 8} = - 0.12 \end{aligned}$

::对数8\\\\ 79=log\\ 79log\\\ 8=-0.12

by Mathispower4u demonstrates how to use the change of base formula to evaluate logarithms with a calculator.
::由 Mathispower4u 演示如何使用基公式的修改来用计算器来评估对数。



How to Evaluate Logarithms With a Scientific Calculator
::如何用科学计算器评估对数

Note: Depending on your calculator, you may need to 1st enter the number and then press LOG or LN to evaluate the logarithm.

How to Evaluate Logarithms With Desmos
::如何用 Desmos 来评估对数

1. Type "log" or "ln."
::1. 型号为“log”或“in”。

2. If your logarithm has a base, use the Underscore key, _ (Shift Hyphen, often next to zero on a keyboard), to enter the base. For fractional bases, it is easier to input them in decimal form.
::2. 如果对数有一个基数,则使用下划线键 _ (键盘上通常不低于零的 Shift 连字符)进入基数,对于分数基数,以小数格式输入它们更容易。

3. Then enter the result and press Enter.
::3. 然后输入结果并按 Enter 键。

How to Evaluate Logarithms With a TI-83/84
::如何用TI-83/84评估对数

1. Press LOG or LN, and then enter the number you are taking the logarithm of in parentheses.
::1. 按 LOG 或 LN 键,然后输入括号中的对数。

2. Press ENTER .
::2. 按ENTER。

Summary
::摘要

To convert between exponential and logarithmic forms, identify the three key parts—the base, the exponent, and the result.
::要在指数表和对数表之间转换,请标明三个关键部分——基、指数和结果。

To evaluate a logarithm without a calculator, consider the relationship in exponential form.
::要评价没有计算器的对数, 请以指数形式考虑此关系。

There are two special logarithms of note —the common log and the natural log.

The common logarithm is the log base 10. We usually write it as $\begin{aligned} \log x, \end{aligned}$ without an explicit base.
::普通对数是对数基数 10, 我们通常把它写成对数基数, 没有明确的基数。

The natural logarithm is the log base $\begin{aligned} e \end{aligned}$ . We write it as $\begin{aligned} \ln x \end{aligned}$ .
::自然对数是日志基数 e. 我们把它写成 INx 。

::备注有两个特殊的对数——共同日志和自然日志。共同的对数是日志10。我们通常在没有明确基数的情况下将其写成logx。自然对数是日志基数,即我们把它写成INx。

To evaluate logarithms using a calculator, you can use the change of base formula, $\begin{aligned} \log_{b} M = \frac{\log_{a} M}{\log_{a} b}, \end{aligned}$ and choose either the common log or the natural log.
::要使用计算器来评估对数, 您可以使用基公式的修改, logbM=logaMlogab, 并选择共同日志或自然日志。

Review
::回顾

Convert the following exponential statements to logarithmic statements:
::将下列指数语句转换为对数语句:

1. $\begin{aligned} 3^{x} = 5 \end{aligned}$
::1. 3x=5

2. $\begin{aligned} a^{x} = b \end{aligned}$
::2.ax=b

3. $\begin{aligned} 4 (5^{x}) = 10 \end{aligned}$
::3. 4(5x)=10

Convert the following logarithmic statements to exponential statements:
::将下列对数语句转换成指数语句:

4. $\begin{aligned} \log_{2} 32 = x \end{aligned}$
::4. log2 32=x

5. $\begin{aligned} \log_{\frac{1}{3}} x = - 2 \end{aligned}$
::5.13x=-2

6. $\begin{aligned} \log_{a} y = b \end{aligned}$
::6.ay=b

Evaluate the following logarithmic expressions without a calculator:
::在无计算器的情况下评价下列对数表达式:

7. $\begin{aligned} \log_{5} 25 \end{aligned}$
::7. log5 25

8. $\begin{aligned} \log_{\frac{1}{3}} 27 \end{aligned}$
::8,1327日志

9. $\begin{aligned} \log \frac{1}{10} \end{aligned}$
::9. log= 110

10. $\begin{aligned} \log_{2} 64 \end{aligned}$
::10.264

11. $\begin{aligned} \ln e^{6} \end{aligned}$
::11. 内 6

12. $\begin{aligned} \log_{5} 1 \end{aligned}$
::12.5%1

Evaluate the logarithmic expressions below using a calculator. You may need to use the change of base formula for some problems.
::使用计算器评估下面的对数表达式。您可能需要对一些问题使用基公式的修改。

13. $\begin{aligned} \ln 8 \end{aligned}$
::13号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号8号

14. $\begin{aligned} \log_{2} 12 \end{aligned}$
::14.212

15. $\begin{aligned} \log_{11} 32 \end{aligned}$
::15. log11_________________________________________________________________________________

Explore More
::探索更多

1. The temperature on the mountain can be measured in kelvins with the formula $\begin{aligned} t = 60 (1 + log (1, 000 - h)), (t < 500) \end{aligned}$ , where h is the height in meters of a given location on the mountain. Approximate the temperature of a location 300 meters up the mountain. Round your answer to the nearest hundredth.
::1. 山上的温度可以用公式t=60(1+log(1 000-h))(t < 500)以海藻测量,其中h是山上某一地点的高度,大约是山上300米的一个地点的温度。

2. The wind velocity in kilometers per hour near the center of a tornado can be measured by $\begin{aligned} v = 92 log d + 65 \end{aligned}$ , where d is the distance in kilometers that the tornado travels. Approximate the wind velocity of a tornado that traveled 100 km. Round your answer to the nearest hundredth.
::2. 龙卷风中心附近每小时每公里的风速可以用 v=92 log d+65 测量,其中d是龙卷风所行行行的公里的距离,大约100公里的龙卷风速度。

by CK-12 Foundation demonstrates how to solve a similar problem to this one.
::CK-12基金会展示了如何解决与此类似的问题。



3. The force in newton that a very physically fit man is able to hold can be measured by $\begin{aligned} f = 500 (1 + log (100 - t)) \end{aligned}$ , where t is the time in seconds that the man has already held. Approximate the force of the man when he has been holding on for 80 seconds. Round your answer to the nearest hundredth.
::3. 一个体格健康的人能够保持的牛顿的威力可以用 f=500(1+log(100-t))来衡量,其中 t 是该男子已经保持的秒数, 接近他保持80秒时的威力, 将你的答复转至最近的100秒。

4. Scientists have found that the length in inches of a certain species of electric eel can be found by $\begin{aligned} L = 22.4 \ln (w - 28.7) \end{aligned}$ , where w is the eel's weight in pounds. Find the weight of an eel that is 48 feet long. Round your answer to the nearest tenth of a pound.
::4. 科学家发现,L=22.4ln(w-28.7)可以找到某种电钢的长度,这里是的重量,找到48英尺长的的重量。

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

References
::参考参考资料

1. "2017 Division 1 Men's Basketball Tournament,"
::1. 2017年第1师男子篮球锦标赛

Section outline

General

评估对数

What Is a Logarithm? ::什么是对数?

Logarithmic Functions ::对对数函数

Converting Between Exponential Form and Logarithmic Form ::对数表和对数表之间的转换

Example 1 ::例1

Example 2 ::例2

Evaluating Logarithms ::评估对数

Example 3 ::例3

WARNING ::警告

Example 4 ::例4

Special Logarithms ::特殊对数

Common Logarithm and Natural Logarithm ::普通对数和自然对数

Example 5 ::例5

Example 6 ::例6

Change of Base Formula ::基基公式变动

Formula ::公式公式公式

Example 7 ::例7

How to Evaluate Logarithms With a Scientific Calculator ::如何用科学计算器评估对数

How to Evaluate Logarithms With Desmos ::如何用 Desmos 来评估对数

How to Evaluate Logarithms With a TI-83/84 ::如何用TI-83/84评估对数

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

References ::参考参考资料

What Is a Logarithm?

::什么是对数?

Logarithmic Functions
::对对数函数

Converting Between Exponential Form and Logarithmic Form
::对数表和对数表之间的转换

Example 1
::例1

Example 2
::例2

Evaluating Logarithms
::评估对数

Example 3
::例3

WARNING
::警告

Example 4
::例4

Special Logarithms
::特殊对数

Common Logarithm and Natural Logarithm
::普通对数和自然对数

Example 5
::例5

Example 6
::例6

Change of Base Formula
::基基公式变动

Formula
::公式公式公式

Example 7
::例7

How to Evaluate Logarithms With a Scientific Calculator
::如何用科学计算器评估对数

How to Evaluate Logarithms With Desmos
::如何用 Desmos 来评估对数

How to Evaluate Logarithms With a TI-83/84
::如何用TI-83/84评估对数

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

References
::参考参考资料