Course: 3.8 普通和自然对数

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共同和自然对数
By now, you know that $\begin{align*}log_2 64=x\end{align*}$ can be solved if you recognize that $\begin{align*}2^{6} = 64\end{align*}$ . What about numbers that aren't so 'clean'? There aren't many people who could calculate the answer to $\begin{align*}log_7 247=x\end{align*}$ in their head! It would be great to use a calculator, but most only have two log functions: base 10 and base e .
::现在, 您知道, log264=x 如果识别到 26=64, 就可以解答 log26=64 。那么不是“ 干净” 的数字呢 ? 没有多少人可以计算他们头上的 log7247=x 的答案。使用计算器会很好, 但大多数只有两个日志功能 : 基数 10 和 base e 。

Is there a way to convert from one base to another, so we can use a calculator?
::有没有办法从一个基地转换到另一个基地, 这样我们就可以使用计算器?

Common and Natural Logarithms
::共同和自然对数

Although a log function can have any positive number as a base, there are really only two bases that are commonly used in the real world. Both may be written without a base noted, like: $\begin{align*}log x\end{align*}$ , so you may need to use the context to decide which is appropriate.
::虽然日志函数可以有一个正数作为基数, 但实际上只有两个基数在现实世界中常用。两种基数都可以在不标注基数的情况下写入, 比如: 日志, 因此您可能需要使用上下文来决定哪个是合适的。

The common log is a log with base 10. It is used to define pH, earthquake magnitude , and sound decibel levels, among many many other common real-world values.
::通用日志是带有10基数的日志,用于界定pH值、地震强度和音响除息水平,以及许多其他共同的现实世界值。

The natural log , sometimes written $\begin{align*}ln (x)\end{align*}$ , is a log with base e . The transcendental number e is approximately 2.71828 and is used in any number of calculations involving constant growth in chemistry, physics, biology, finance, etc..
::自然日志,有时是In(x)书写的,是带有基数的日志。超数约为2.71828,用于化学、物理学、生物学、生物学、金融等任何数量涉及化学、物理、生物学、金融等持续增长的计算。

Using a Calculator for logs
::使用计算日志的计算器

You may have noticed that the common log and the natural log are the only log buttons on your calculator. We can use either the common log or the natural log to find the values of logs with other bases.
::您可能已经注意到, 普通日志和自然日志是计算器上唯一的日志按钮。我们可以使用普通日志或自然日志来用其他基准查找日志的值。

The equation $\begin{align*}log_b x = \frac{log x} {log b}\end{align*}$ is called the formula, and may be used to convert to common log or natural log .
::方程式logbx=logxlogb 被称为公式,可用于转换成共同日志或自然日志。

You may also see the change of base formula as $\begin{align*}\log_b x = \frac{ln x} {ln b}\end{align*}$ , which is the same formula specifying a conversion to the natural log .
::您也可以看到基数公式的修改为 logbx=lnxlnb, 即指定转换为自然日志的相同公式。

Using the change of base formula, we can find the common log (or the natural log) equivalent of any other base so that we can use a calculator to find the value of an expression.
::使用基数公式的改变,我们可以找到任何其他基数的普通日志(或自然日志)等值,以便我们可以使用计算器找到表达式的值。

Consider log ₃ 35. If we use the change of base formula to convert to base 10, and then the $\begin{align*}log\end{align*}$ button on a calculator, we find that $\begin{align*}log_3 35 = \frac{log35} {log3} = 3.23621727\end{align*}$ .
::35. 如果我们使用基公式的改变转换为基数 10, 然后在计算器上按下日志按钮, 我们就会发现对数 335=log35log3=3.23621727。

Examples
::实例

Example 1
::例1

Earlier, you were asked to solve the following problem: $\begin{align*}log_7 247=x\end{align*}$ .
::早些时候,你被要求解决以下问题:log7247=x。

Using the change of base formula: $\begin{align*}log_7 247 = \frac{log 247} {log 7}\end{align*}$ .
::使用基公式的更改:log7247=log247log7。

Using a calculator to find the common logs of 247 and 7, we get (approximately) $\begin{align*}\frac{2.4} {.8} = 2.8313\end{align*}$ .
::使用计算器找到247和7的常见日志,我们得到(约)2.4.8=2.8313。

We can verify with $\begin{align*}7^{2.8313} = 247\end{align*}$
::我们可以用72 8313=247来验证

$\begin{align*}\therefore log_7 247 = 2.8313\end{align*}$
::@log7247=2.8313

Example 2
::例2

Evaluate each log.
::评估每个日志。

Remember that $\begin{align*}log x\end{align*}$ (with no base specified) commonly refers to $\begin{align*}log_{10} x\end{align*}$ .
::记住日志( 没有指定基数) 通常指日志10x 。

$\begin{align*}log 1\end{align*}$
::日志1

$\begin{align*}log 1 = 0\end{align*}$ because $\begin{align*}10^{0} = 1\end{align*}$ .
::log1=0 因为 100=1

$\begin{align*}log 10\end{align*}$
::日志10

$\begin{align*}log 10 = 1\end{align*}$ because $\begin{align*}10^{1} = 10\end{align*}$
::log10=1 因为 101=10

$\begin{align*}log\sqrt{10}\end{align*}$
::log_________________________________________________________________________________________________________________________

$\begin{align*}log \sqrt{10} = \frac{1} {2}\end{align*}$ because $\begin{align*}\sqrt{10} = 10^{1/2}\end{align*}$
::log_10=12 因为10=101/2

Example 3
::例3

For each log value, determine two integers between which the log value should lie. Then use a calculator to find the value of the log.
::对于每个日志值, 确定日志值之间应该包含的两个整数。然后使用计算器查找日志值。

log 50
::50 log 50

The value of this log should be between 1 and 2, as 10 ¹ = 10, and 10 ² = 100.
::该日志值应在1至2之间,101=10,102=100。

Using a calculator, you should find that log 50 ≈ 1.698970004.
::使用计算器,你应该找到对数 50 1.69897004。

log 818
::log 818

The value of this log should be between 2 and 3, as 10 ² = 100, and 10 ³ = 1000.
::该日志值应在2至3之间,即102=100,103=1000。

Using a calculator, you should find that log 818 ≈ 2.912753304.
::使用计算器,你应该找到818 2.912753304的日志。

Example 4
::例4

Estimate the value, and then use the change of base formula to find the value of $\begin{align*}log_2 17\end{align*}$ .
::估计值,然后使用基公式的修改来查找log217的值。

$\begin{align*}log_2 17\end{align*}$ is close to 4 because $\begin{align*}2^{4} = 16\end{align*}$ and $\begin{align*}2^{5} = 32\end{align*}$ .
::log217 接近 4, 因为 24 = 16 和 25 = 32 。

Using the change of base formula, we have $\begin{align*}log_2 17 = \frac{log 17} {log 2}\end{align*}$ .
::使用基数公式的更改,我们找到对数217=log17log2。

Using a calculator, you should find that the approximate value of this expression is 4.087462841.
::使用计算器,您应该发现这个表达式的大约值是4.087462841。

Example 5
::例5

Find the value of each natural log.
::查找每个自然日志的值。

$\begin{align*}ln 100\end{align*}$
::内内 100

$\begin{align*}ln 100\end{align*}$ is between 4 and 5. You can estimate this by rounding $\begin{align*}e\end{align*}$ up to 3, and considering powers of 3:
::In100在4到5之间你可以通过四舍五入 e 到 3 来估计并且考虑 3 的权力 3 :

$\begin{align*}3^4 = 81\end{align*}$ and $\begin{align*}3^5 = 243\end{align*}$
::34=81和35=243

Using a calculator, you should find that $\begin{align*}ln 100\end{align*}$ ≈ 4.605171086.
::使用计算器,你应该发现 In100 4.605171086。

$\begin{align*}ln \sqrt{e}\end{align*}$
::内内

Recall that a square root is the same as an exponent of 1/2.
::回顾平方根与1/2的标语相同。

Therefore $\begin{align*}ln \sqrt{e} = ln (e^{1/2}) = 1/2\end{align*}$ .
::因此Ine=ln(e1/2)=1/2。

Example 6
::例6

Solve the equation: $\begin{align*}3^x(2^{3x}) = 7(5^x)\end{align*}$
::解析方程: 3x( 23x) = 7( 5x)

To solve:
::要解决 :

$\begin{align*}3^x(2^3)^x = 7(5^x)\end{align*}$ : Rule of exponents $\begin{align*}(x^y)^z = x^{yz}\end{align*}$
::3x( 23)x=7( 5x) : exponents (xy)z=xyz 规则

$\begin{align*}3^x(8^x) = 7(5^x) \to 24^x = 7(5^x)\end{align*}$ : By multiplication
::3x( 8x) = 7( 5x) = 24x= 7( 5x) : 乘以乘法

$\begin{align*}(\frac{24}{5})^x = 7\end{align*}$ : Divide both sides by $\begin{align*}5^x\end{align*}$
:245x=7):双方除以5x

$\begin{align*}log (\frac{24}{5})^x = log 7\end{align*}$ : Take the log of both sides
::log( 245x) =log7 : 使用两侧的日志

$\begin{align*}xlog (\frac{24}{5}) = log 7\end{align*}$ : Using $\begin{align*}log x^y = y log x\end{align*}$
::xlog( 245) =log7 : 使用对数=ylogx

$\begin{align*}x = \frac{log7}{log\frac{24}{5}}\end{align*}$ : Divide both sides by $\begin{align*}log (\frac{24}{5})\end{align*}$
::x=log7log245:将两边除以对数( 245)

$\begin{align*}x = 1.24\end{align*}$ : With a calculator
::x=1.24 : 使用计算器

Example 7
::例7

Find the value: $\begin{align*}ln6 + ln7\end{align*}$ .
::查找值: in6+ln7。

Use a calculator to find the values:
::使用计算器查找值 :

$\begin{align*}ln6 = 1.79175\end{align*}$ and $\begin{align*}ln7 = 1.94591\end{align*}$
::In6=1.79175和In7=1.94591

$\begin{align*}1.79175 + 1.94591 = 3.73766\end{align*}$

Review
::回顾

What is a common logarithm? Where are common logs most commonly used?
::常见对数是什么? 最常用的普通日志在哪里 ?

What is a natural logarithm? Where are natural logs commonly used?
::天然对数是什么? 天然对数在哪里常用? 天然对数在哪里 ?

Evaluate each expression:
::评估每个表达式:

$\begin{align*}log\frac{17^4}{5}\end{align*}$
::日志1745

$\begin{align*}log 7(4^3)\end{align*}$
::log7(43)

Convert to a common logarithm and evaluate:
::转换为通用对数并评价 :

$\begin{align*}log_{6} 832\end{align*}$
::日志6832

$\begin{align*}log_{11} 47\end{align*}$
::日志1147

$\begin{align*}log_{3} 9\end{align*}$
::log39

Convert to a natural logarithm and evaluate:
::转换为自然对数并评估 :

$\begin{align*}log_{7} 94\end{align*}$
::log794 log794

$\begin{align*}log_{5} 256\end{align*}$
::日志5256

$\begin{align*}log_{9} 0.712\end{align*}$
::日志90.712

Find the values of the natural logarithms:
::查找自然对数的值 :

$\begin{align*}ln 56\end{align*}$
::5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

$\begin{align*}ln 2000\end{align*}$
::2000年

$\begin{align*}ln 950.1\end{align*}$
::In950.1

$\begin{align*}ln 0.9\end{align*}$
::In0. 9 中

Convert the natural logs to exponential form, and solve.
::将自然日志转换成指数形式并解析。

If $\begin{align*}ln e = x\end{align*}$ and $\begin{align*}e^x = e\end{align*}$ then $\begin{align*}x =?\end{align*}$
::如果 Ine=x 和 ex=e 那么 x=?

If $\begin{align*}ln e^5=x\end{align*}$ then $\begin{align*}x =?\end{align*}$
::如果 Ine5=x 那么 x=?

If $\begin{align*}ln e^a = x\end{align*}$ then $\begin{align*}x =?\end{align*}$
::如果 Inea=x 那么 x=?

If $\begin{align*}ln e^{-3} = x\end{align*}$ then $\begin{align*}x =?\end{align*}$
::如果 Ine - 3=x 那么 x=?

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

共同和自然对数

Common and Natural Logarithms ::共同和自然对数

Using a Calculator for logs ::使用计算日志的计算器

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Example 7 ::例7

Review ::回顾

Review (Answers) ::回顾(答复)

Common and Natural Logarithms
::共同和自然对数

Using a Calculator for logs
::使用计算日志的计算器

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Example 7
::例7

Review
::回顾

Review (Answers)
::回顾(答复)