Course: 3.5 基数变动 | The Way To Learn

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基数变动

While it is possible to change bases by always going back to exponential form, it is more efficient to find out how to change the base of logarithms in general. Since there are only base

\begin{aligned} e \end{aligned}

and base 10 logarithms on most calculators, how would you evaluate an expression like

\begin{aligned} \log_{3} 12 \end{aligned}

Changing the Base of Logarithms
::更改对数基

The change of base property states:
::基本财产的变更规定如下:

$\begin{aligned} \log_{b} x = \frac{\log_{a} x}{\log_{a} b} \end{aligned}$
::对数bx=logaxlogab

You can derive this formula by converting $\begin{aligned} \log_{b} x \end{aligned}$ to exponential form and then taking the log base $\begin{aligned} x \end{aligned}$ of both sides. This is shown below.
::您可以将logbx转换成指数形式,然后取出两边的对数基数 x 来得出此公式。下面将显示这一点。

\begin{aligned} \log_{b} x & = y \\ b^{y} & = x \\ \log_{a} b^{y} & = \log_{a} x \\ y \log_{a} b & = \log_{a} x \\ y & = \frac{\log_{a} x}{\log_{a} b} \end{aligned}

::logb_x=yby=xloga_by=loga_xyloga_b=loga_xyloga_xy=loga_xloga_b =loga_xloga_xloga_b =loga_xloga_b loga_xloga_xloga_b loga_b =loga_xloga_xloga_b loga_b=loga_xy=loga_xloga_xy_b loga_xy_xy=loga_xy=loga_xloga_b loga=loga_xloga_xloga_b=loga_b loga_x_xx_xy=loga_xloga_b=loga________________________________________________________________________________

Therefore, $\begin{aligned} \log_{b} x = \frac{\log_{a} x}{\log_{a} b} \end{aligned}$ .
::因此,logb*x=loga*xloga*b。

If you were to evaluate $\begin{aligned} \log_{3} 4 \end{aligned}$ using your calculator, you may need to use the change of base formula since some calculators only have base 10 or base $\begin{aligned} e \end{aligned}$ . The result would be:
::如果您使用计算器对日志34进行评估,可能需要使用基数公式的修改,因为有些计算器只有基数 10 或基数 e 。结果将是:

$\begin{aligned} \log_{3} 4 = \frac{\log_{10} 4}{\log_{10} 3} = \frac{\ln 4}{\ln 3} \approx 1.262 \end{aligned}$
::34=log104log103=n4ln}31.262

Examples
::实例

Example 1
::例1

Earlier, you were asked how to use a calculator to evaluate an expression like $\begin{aligned} \log_{3} 12 \end{aligned}$ . In order to evaluate an expression like $\begin{aligned} \log_{3} 12 \end{aligned}$ you have some options on your calculator:
::早些时候, 有人询问您如何使用计算器来评价像 log3\\\ 12 这样的表达式。为了评估像 log3\\ 12 这样的表达式, 您在计算器上有一些选项 :

$\begin{aligned} \frac{\ln 12}{\ln 3} = \frac{\log 12}{\log 3} \approx 2.26 \end{aligned}$
::======================================================================================================================================================================================================================================== ====================================================================================================================================================================================================================================================================================

Some graphing calculators also have another option. Press the MATH followed by the A buttons and enter $\begin{aligned} \log_{3} 12 \end{aligned}$ .
::一些图形计算器还有另一个选项。按 A 按钮之后的 MATH 键并输入对数 3\\\\\ 12 。

Example 2
::例2

Prove the following log identity.
::证明以下日志身份。

$\begin{aligned} \log_{a} b = \frac{1}{\log_{b} a} \end{aligned}$
::对数ab=1logba

$\begin{aligned} \log_{a} b = \frac{\log_{x} b}{\log_{x} a} = \frac{1}{\frac{\log_{x} a}{\log_{x} b}} = \frac{1}{\log_{b} a} \end{aligned}$
::loga_b=logx_blogx_a=1logx_alogx_b=1logb=1logb_a

Example 3
::例3

Simplify to an exact result: $\begin{aligned} (\log_{4} 5) \cdot (\log_{3} 4) \cdot (\log_{5} 81) \cdot (\log_{5} 25) \end{aligned}$
::简化到一个确切结果log45) (log34) (log581) (log525)

$\begin{aligned} \frac{\log 5}{\log 4} \cdot \frac{\log 4}{\log 3} \cdot \frac{\log 3^{4}}{\log 5} \cdot \frac{\log 5^{2}}{\log 5} = \frac{\log 5}{\log 4} \cdot \frac{\log 4}{\log 3} \cdot \frac{4 \cdot \log 3}{\log 5} \cdot \frac{2 \cdot \log 5}{\log 5} = 4 \cdot 2 = 8 \end{aligned}$
::

Example 4
::例4

Evaluate: $\begin{aligned} \log_{2} 48 - \log_{4} 36 \end{aligned}$
::评价: log248-log436

\begin{aligned} \log_{2} 48 - \log_{4} 36 & = \frac{\log 48}{\log 2} - \frac{\log 36}{\log 4} \\ = \frac{\log 48}{\log 2} - \frac{\log 6^{2}}{\log 2^{2}} \\ = \frac{\log 48}{\log 2} - \frac{2 \cdot \log 6}{2 \cdot \log 2} \\ = \frac{\log 48 - \log 6}{\log 2} \\ = \frac{\log (\frac{48}{6})}{\log 2} \\ = \frac{\log 8}{\log 2} \\ = \frac{\log 2^{3}}{\log 2} \\ = \frac{3 \cdot \log 2}{\log 2} \\ = 3 \end{aligned}

::_____________________________________________________23________________________________3________3_____

Example 5
::例5

Given $\begin{aligned} \log_{3} 5 \approx 1.465 \end{aligned}$ find $\begin{aligned} \log_{25} 27 \end{aligned}$ without using a log button on the calculator.
::给定对数 351.465 找到对数 2527 没有在计算器上使用日志按钮。

$\begin{aligned} \log_{25} 27 = \frac{\log 3^{3}}{\log 5^{2}} = \frac{3}{2} \cdot \frac{1}{(\frac{\log 5}{\log 3})} = \frac{3}{2} \cdot \frac{1}{\log_{3} 5} \approx \frac{3}{2} \cdot \frac{1}{1.465} = 1.0239 \end{aligned}$
::log25\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Summary

The change of base property states: $\begin{aligned} \log_{b} x = \frac{\log_{a} x}{\log_{a} b} \end{aligned}$ for any $\begin{aligned} a . \end{aligned}$
::基本财产的变更表示:a.

Using the change of base formula, you can evaluate expressions that have less common bases with a calculator that only has base 10 or base $\begin{aligned} e \end{aligned}$ logarithms.
::使用基公式的修改,您可以使用一个只有基数 10 或基数 e 对数的计算器,对基础值较少的表达式进行评价。

Review
::回顾

Evaluate each expression by changing the base and using your calculator.
::通过改变基数和使用计算器来评价每个表达式。

1. $\begin{aligned} \log_{6} 15 \end{aligned}$
::1. 日数615

2. $\begin{aligned} \log_{9} 12 \end{aligned}$
::2. 日对数912

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

Evaluate each expression.
::评估每个表达方式。

4. $\begin{aligned} \log_{8} (\log_{4} (\log_{3} 81)) \end{aligned}$
::4.8(log4(log381))

5. $\begin{aligned} \log_{2} 3 \cdot \log_{3} 4 \cdot \log_{6} 16 \cdot \log_{4} 6 \end{aligned}$
::5.23344616466666

6. $\begin{aligned} \log 125 \cdot \log_{9} 4 \cdot \log_{4} 81 \cdot \log_{5} 10 \end{aligned}$
::6. log *==================================================================================================================================

7. $\begin{aligned} \log_{5} (5^{\log_{5} 125}) \end{aligned}$
::7.5(5log5125)

8. $\begin{aligned} \log (\log_{6} (\log_{2} 64)) \end{aligned}$
::8.(log6(log264))

9. $\begin{aligned} 10^{\log_{100} 9} \end{aligned}$
::9. 10log1009

10. $\begin{aligned} (\log_{4} x) (\log_{x} 16) \end{aligned}$
::10. (log4x)(logx}16)

11. $\begin{aligned} \log_{49} 49^{5} \end{aligned}$
::11.4949*495

12. $\begin{aligned} 3 \log_{24} 24^{8} \end{aligned}$
::12. 3log24 248

13. $\begin{aligned} 4^{\log_{2} 3} \end{aligned}$
::13. 4log2%3

Prove the following properties of logarithms.
::证明对数的下列属性。

14. $\begin{aligned} (\log_{a} b) (\log_{b} c) = \log_{a} c \end{aligned}$
::14. (loga_b)(logb_c)=loga_c (loga_c) =loga_c (loga_b)(logb_c) =loga_c (loga_c) =loga_c (loga_b)(loga_b)(logb_c) =loga_c (loga_c) =loga_c (loga_c)

15. $\begin{aligned} (\log_{a} b) (\log_{b} c) (\log_{c} d) = \log_{a} d \end{aligned}$
::15. (logab)(logbc)(logcd) =logad (logab)(logbc)(logcd) =logad)

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

基数变动

Changing the Base of Logarithms ::更改对数基

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)