课程： 5.5 对数函数-interactive

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对对数函数
Lesson Objectives
::经验教训目标

Find the inverse of a logarithmic function.
::查找对数函数的反向。

Find the inverse of a natural logarithmic function.
::查找自然对数函数的反向值。

Introduction: The Weber-Fechner Law
::导言:Weber-Fechner法

Use the interactive below to explore the concept behind the work of psychologists Ernst Heinrich Weber and Gustav Fechner.
::利用以下互动来探讨心理学家Ernst Heinrich Weber和Gustav Fechner的工作背后的概念。

Ernst Heinrich Weber explored, and Gustav Fechner advanced, the idea that changes become less noticeable at higher intensities. This idea is useful in a wide range of areas. In business, for example, marketing strategists need to have a strong understanding of this relationship when raising prices. Price changes are used to increase revenue, but an increase that is perceived as too large could decrease revenue.
::恩斯特·海因里希·韦伯(Ernst Heinrich Weber)和古斯塔夫·费克纳(Gustav Fechner)探讨了在较高强度下变化不那么明显的想法。这一想法在一系列广泛领域都有用。比如,在商业领域,营销战略家在提高价格时需要深入了解这种关系。价格变化被用来增加收入,但认为增长太大可能会减少收入。

This concept is important to logarithms because the graph produced by the interactive is a logarithmic function. The interactive above highlights the way your brains perceive change logarithmically. You may have noticed that time seems to pass faster with age. One year to a four-year-old represents 25% of his or her life experience. However, one year to a 50-year-old represents 2% of his or her life experience.
::这个概念对于对数很重要, 因为互动生成的图表是一个对数函数。上面的交互性突出显示您大脑对变化的认知方式。您可能注意到时间随着年龄的变迁而过得更快。一岁到四岁代表其生活经历的25%。然而, 一岁到五十岁代表其生活经历的2% 。

T he remainder of the chapter will explore logarithmic functions, how they are derived, and their uses in society.
::本章的其余部分将探讨对数功能、如何产生这些功能及其在社会中的用途。

Activity 1: Inverse Functions Revisited
::活动1:重新审视反向职能

R ecall that the inverse of a function can be graphed by reflecting it over the line formed by the linear equation $\begin{align*}y = x.\end{align*}$
::回顾函数的反面可以通过在由线性方程式y=x构成的线条上反射来图解函数。

Example
::示例示例示例示例

Graph the set of points $\begin{align*}\{(1,0), (3,0), (3,2)\}\end{align*}$ and graph the inverse of this set.
::绘制 {(1,0,(3,),(3,),(3,2)} 的点数图,并绘制本集的反方向图。

Let's start by g raphing the line $\begin{align*}y=x\end{align*}$ on a coordinate plane, as well as the points $\begin{align*}(1,0), (3,0),\end{align*}$ and $\begin{align*}(3,2).\end{align*}$ Now reflect these three coordinates over the line, and draw/label the new reflected points. Recall that the inverse of a coordinate is the reverse of the $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ values. The inverse of the point $\begin{align*}(x,y)\end{align*}$ is $\begin{align*}(y,x).\end{align*}$ The graph below shows the original points in red and the reflected points in purple. Notice that the original coordinates reflected over the line $\begin{align*}y = x\end{align*}$ result in new coordinates that are equidistant from the line that they are reflected over.
::我们首先用图解坐标平面上的线y=x以及点(1,0,(3,),(3,)和(3,2)来绘制坐标,然后在线上绘制这三个坐标,然后绘制/标出新的反映点。回顾一个坐标的反方向是x和y值的反方向。点的反方向(x,y)是(y,x)。下图显示红色的原点和紫色的反向点。请注意,原坐标反映于线y=x之后的新坐标与反向线的坐标相等。

Answer:
::答复:

Points reflected across the line y=x
::y=x 横线反射的点数

Activity 2: Deriving Logarithmic Functions
::活动2:产生对数函数

A logarithm represents the exponent to which a base is raised to produce a product . Additionally, a logarithm is the inverse of an exponent.
::对数表示用于生产产品的基数。此外,对数是表数的反数。

Knowing that logarithms are inverses of exponents, you can infer that the logarithmic function will be an inverse of an exponential function with the same base.
::明知对数是引数的反函数,可以推断对数函数是同一个基数的指数函数的反函数。

Example
::示例示例示例示例

Write the inverse of the function $\begin{align*}f(x)=5^x.\end{align*}$
::写入函数 f(x)=5x 的反数。

In , you learned that the following steps are necessary to write the inverse of a function :
::在其中,你了解到有必要采取以下步骤来撰写一个函数的反面文字:

1. Switch $\begin{align*}x\end{align*}$ and $\begin{align*}y.\end{align*}$
::1. 交换x和y。

2. Solve for $\begin{align*}y.\end{align*}$
::2. 解决y.

Let's apply these steps to the equation $\begin{align*}y=5^x.\end{align*}$
::让我们将这些步骤应用到 y=5x 等式上。

1. Switch $\begin{align*}x\end{align*}$ and $\begin{align*}y.\end{align*}$
::1. 交换x和y。

$\begin{aligned} y = 5^{x} \to x = 5^{y} \end{aligned}$
$\begin{align*}y = {5}^{\:x} → x = {5}^{\:y}\end{align*}$
::y=5xxx=5y

2. Solve for $\begin{align*}y.\end{align*}$
::2. 解决y.

This step is a little more complicated. To solve for $\begin{align*}y,\end{align*}$ use the inverse of an exponent. Luckily, you already know that the inverse of an exponent is a logarithm.
::这个步骤比较复杂。要解决y, 请使用推论方的反面。幸运的是, 您已经知道推论方的反面是对数。

$\begin{aligned} x = 5^{y} \to y = \log_{5} x \end{aligned}$
$\begin{align*}x = {5}^{\:y} → y = \log_{5}{x}\end{align*}$
::x=5y=log5x

Writing this equation in results in the answer.
::将这个方程式写成答案。

Answer: $\begin{align*}f(x) = \log_{5}{x}\end{align*}$
::答复:f(x)=log5_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

T ake a deeper look at how converting an exponential equation to its logarithmic form. Even though a logarithm represents an exponent, it can be applied to both sides of an equation.
::更深入地查看指数方程式如何转换成对数形式。即使对数代表了引号, 它也可以适用于方程式的两侧。

$\begin{aligned} x = 5^{y} \to \log_{5} (x) = \log_{5} (5^{y}) \to \log_{5} x = y \end{aligned}$
$\begin{align*}x = {5}^{\:y}→\color{blue}{\log_{5}{(\color{black}{x}\color{blue}{)}}} \color{black}~{ = }~ \color{blue}{\log_{5}{(\color{black}{5^y}\color{blue}{)}}}\color{black}{→ \log_{5}{x} = y}\end{align*}$
::x=5ylog5(x) = log5(5y) = log5(x) = log5(x) y

A pplying the base 5 log to the left side will produce $\begin{align*}\log_{5}{x}\end{align*}$ as expected. However, an operation to the left side of an equation cannot be performed without performing the same operation to the right. A pplying the base 5 log to the right side of the equation will result in $\begin{align*}\log_{5}{5^y}.\end{align*}$ Recall that this is asking the following, "5 raised to what power is $\begin{align*}5^y\end{align*}$ ?" C onsider a few examples to help illustrate this.
::将基数 5 日志应用到左侧将产生预期的对log5 =%x 。但是, 如果对等方程左侧的操作不执行对右侧的相同操作, 则无法进行对等方程左侧的操作。将基数 5 日志应用到对等方程右侧将导致对log5 =5 =5 y 。回顾这是在询问下面的问题 , “ 5 升到5 y 的功率是多少? ” 请举几个例子来说明这一点。

$\begin{align*}\text{If } 5^x = 5^2, x = 2.\end{align*}$
::如果5x=52,x=2,则5x=52,x=2。

$\begin{align*}\text{If } 5^x = 5^3, x = 3.\end{align*}$
::如果5x=53,x=3,则5x=53,x=3。

$\begin{align*}\text{If } 5^x = 5^4, x = 4.\end{align*}$
::如果5x=54x=4的话

$\begin{align*}\text{If } 5^x = 5^y, \text{ then }x = y.\end{align*}$
::如果5x=5y,则x=y。

U se this approach to derive the logarithmic function for any base.
::使用此方法得出任何基数的对数函数。

$\begin{aligned} y = \log_{b} x \end{aligned}$
$\begin{align*}y = \log_{b}{x}\end{align*}$
::y=logb_x y=logb_x y=logb_x y=logb_x y=logb_x

Here's how this function would look in exponential form :
::以下是这个函数以指数形式呈现的方式 :

$\begin{aligned} b^{y} = x \end{aligned}$
$\begin{align*}b^y = x\end{align*}$
::结束时间=x

Answer the questions below to explore this concept further.
::回答以下进一步探讨这一概念的问题。

Discussion Question : What strategies did you use to solve for the inverse of exponential equations that were more complex than the standard $\begin{align*}y = b^x?\end{align*}$
::讨论问题:对于比标准y=bx更复杂的反指数方程式,你用什么战略来解决这些反指数方程式?

Activity 3: Graphing Logarithmic Functions
::活动3:绘制对数函数图

Using what you know about inverses, u se the inverse of a logarithmic function to help graph it.
::使用对数函数的反对数来帮助绘制图。

Example
::示例示例示例示例

Use the inverse to help g raph the logarithmic function $\begin{align*}f(x) = \log_{2}{(\frac{1}{3}x)}.\end{align*}$
::使用反向来帮助绘制对数函数 f( x) =log2\\\\( 13x) 的图解。

The inverse of this function is $\begin{align*}f^{-1}(x) = 3\cdot2^x ,\end{align*}$ which you already know how to graph. Use the interactive below to finish this example.
::此函数的逆数是 f-1( x) =32x, 您已经知道如何绘制图形。使用下面的交互效果来完成此示例。

Use the interactive below to further explore the relationship between a logarithmic function and its inverse exponential function.
::使用下面的交互功能来进一步探索对数函数与其反指数函数之间的关系。

Activity 4: Natural Log Functions
::活动4:自然日志函数

In the previous section, Logarithms, you learned that the logarithm with a base of $\begin{align*}e\end{align*}$ is called the natural logarithm.
::在上一节 " Logarithms " 中,你了解到e基对数的对数称为自然对数。

The Natural Logarithm
::自然对数

$\begin{align*}\log_{e}{x}=\ln{x}\end{align*}$

G raph the natural logarithm the same way as a logarithm with any base. Answer the questions below to explore graphs of the natural log.
::以任何基点的对数相同的方式绘制自然对数。回答下面的问题, 以探索自然对数的图表。

Wrap-Up: Review Questions
::总结:审查问题

Summary
::摘要

An i nverse function switches the output and input of a function.
::反向函数切换函数的输出和输入。

When finding the inverse of a function, replace the $\begin{align*}x'\end{align*}$ s with y and replace the $\begin{align*}y'\end{align*}$ s with an $\begin{align*}x,\end{align*}$ then solve for $\begin{align*}y.\end{align*}$
::当查找函数的反义时,用 y 替换 x 和 y 替换 y ,然后为 y 解决 y 。

$\begin{align*}b^x=a\end{align*}$ can be written as $\begin{align*}\log_b a=x.\end{align*}$
::bx=a 可以用logba=x 写入。

The natural logarithm is in the form $\begin{align*}\ln x.\end{align*}$ Note that $\begin{align*}\log_e x=\ln x.\end{align*}$
::自然对数以 INx 的形式出现。注意 logex=lnx 。

章节大纲

常规

对对数函数

Introduction: The Weber-Fechner Law ::导言:Weber-Fechner法

Activity 1: Inverse Functions Revisited ::活动1:重新审视反向职能

Activity 2: Deriving Logarithmic Functions ::活动2:产生对数函数

Activity 3: Graphing Logarithmic Functions ::活动3:绘制对数函数图

Activity 4: Natural Log Functions ::活动4:自然日志函数

The Natural Logarithm ::自然对数

Wrap-Up: Review Questions ::总结:审查问题

Summary ::摘要

Introduction: The Weber-Fechner Law
::导言:Weber-Fechner法

Activity 1: Inverse Functions Revisited
::活动1:重新审视反向职能

Activity 2: Deriving Logarithmic Functions
::活动2:产生对数函数

Activity 3: Graphing Logarithmic Functions
::活动3:绘制对数函数图

Activity 4: Natural Log Functions
::活动4:自然日志函数

The Natural Logarithm
::自然对数

Wrap-Up: Review Questions
::总结:审查问题

Summary
::摘要