摘要: 指数函数和对数函数

In this chapter, we learned about: 
::在本章中,我们了解到:

One-to-One Functions and Inverse Functions
::一对一函数和反函数

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• A function is one-to-one if $\begin{array}{r}f(x_1)=f(x_2)\end{array}$  implies $\begin{array}{r}{x}_1=x_2\end{array}$ .
  ::如果 f( x1) =f( x2) 意味着 x1=x2, 一个函数为一对一。
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• We can see that a function is one-to-one by using the horizontal line test: If every horizontal line drawn through the graph intersects it at most once, then the function is one-to-one.
  ::通过水平线测试,我们可以看到一个函数是一对一:如果通过图形绘制的每一水平线最多一次交叉,则函数是一对一。
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• The domain of $\begin{array}{r}f\end{array}$  is the range of $\begin{array}{r}{f}^{\text{-}1}\end{array}$  and the range of $\begin{array}{r}f\end{array}$  is the domain of $\begin{array}{r}{f}^{\text{-}1}\end{array}$ .
  ::f 的域是 f-1 的域, f 的域是 f-1 的域。
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• Functions and their inverses are symmetric about the line $\begin{array}{r}y=x\end{array}$ .
  ::函数及其反函数与 y=x 线对称。
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• To find an inverse function algebraically, determine that the function is one-to-one, switch the variables for the input and the output (usually $\begin{array}{r}x\end{array}$  for the input and $\begin{array}{r}y\end{array}$  for the output), and solve for the output variable.
  ::要找到反向函数代数,确定函数为一对一,切换输入和输出的变量(输入和输出通常为x,输出为y),并解析输出变量。
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• To check inverse functions, compose the function and its inverse twice to see if the result is $\begin{array}{r}x\end{array}$ —that is, $\begin{array}{r}(f\circ f^{\text{-}1})(x)=x\end{array}$  and $\begin{array}{r}(f^{\text{-}1}\circ f)(x)=x\end{array}$ . If a function is not one-to-one, restrict its domain to a section of the graph that is one-to-one.
  ::要检查反向函数,请将函数和反向函数组成两次以查看结果是否为 x - 即 (ff-1)(x)=x 和 (f-1f)(x)=x。如果函数不是一对一,请将其域限制在图中的一对一部分。

Exponential Functions and the Number e
::指数函数和数字e

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• Data in a table of values is exponential if there is a factor by which you can multiply or divide to get the other outputs.
  ::数值表中的数据是指数,如果有一个系数可以乘或除以获得其他输出的话。
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• An exponential function is a function of the form $\begin{array}{r}y=a{b}^x,\end{array}$  where $\begin{array}{r}b>0,b\ne 1\end{array}$  and $\begin{array}{r}a\ne 0\end{array}$  is a real number.
  ::指数函数为y=abx形式的函数,其中 b>0,b1和a0是一个实际数字。
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• If $\begin{array}{r}b>1\end{array}$ , the graph will increase from left to right, and the function is an exponential growth function. If $\begin{array}{r}0<b<1\end{array}$ , the graph will decrease from left to right, and the function is an exponential decay function.
  ::如果 b>1, 图形将从左向右增长, 函数是一个指数增长函数。 如果 0<b < 1, 图形将从左向右下降, 函数是一个指数衰减函数 。
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• To find an exponential function given points, substitute the known values into $\begin{array}{r}y=a{b}^x\end{array}$  and solve the system of equations.
  ::要找到指数函数给定点, 请将已知值替换为 y=abx 并解析方程系统 。
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• To graph an exponential function, make a table of values.
  ::要绘制指数函数图,请绘制一个数值表。
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• The number $\begin{array}{r}e\end{array}$  is approximately 2.72.
  ::e 数字约为2.72。
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• To graph a function of the form $\begin{array}{r}y=e^x,\end{array}$  approximate it with 3.
  ::要绘制表y=ex 的函数,请将其与 3 相近。

Evaluating Logarithms and Logarithmic Functions
::评估对数和对数函数

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• 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。
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• 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 。
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• There are two special logarithms of note—the common log and the natural log.
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  • The common logarithm is the log base 10. We usually write is as $\begin{array}{r}\log x\end{array}$  without an explicit base.
    ::通用对数是对数基数 10。 我们通常写为对数基数, 没有明确的基数 。
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  • The natural logarithm is the log base e. We write it as $\begin{array}{r}\ln x\end{array}$ .
    ::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10对数基数。我们通常在没有明确基数的情况下写成对数基数。自然对数基数是日数基数。我们把它写成INQx。
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• To evaluate logarithms using a calculator, you can use the change of base formula, $\begin{array}{r}{\log }_{b}x=\frac{{\log }_{a}x}{{\log }_{a}b},\end{array}$  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, logbx=logaxlogab, 并选择共同日志或自然日志 。
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• To find the domain of a logarithmic function, set the result greater than 0 and solve. The values that are part of the solution are the domain of the function.
  ::要查找对数函数的域,请将结果设定为大于 0 并解析。作为解决方案一部分的值是函数的域。
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• To graph a logarithmic function, find the domain, make a table of values with three values, draw the vertical asymptote if it is not one of the axes, and graph the curve.
  ::要绘制对数函数图,请查找域,绘制三个值的数值表,如果不是轴之一,则绘制垂直零点图,并绘制曲线图。

Simplifying and Expanding Logarithms Using Logarithmic Properties
::使用对数属性简化和扩展对数属性

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• The logarithmic properties are:
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  • Zero Exponent:  $\begin{array}{r}{\log }_{b}1=0\end{array}$  
    ::零指数: logb1=0
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  • Identity:  $\begin{array}{r}{\log }_{b}b=1\end{array}$  
    ::身份: logbb=1
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  • Product Rule:  $\begin{array}{r}{\log }_b(xy)={\log }_{b}x+{\log }_{b}y\end{array}$  
    ::产品规则: logb( xy) =logbx+logby
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  • Quotient Rule:  $\begin{array}{r}{\log }_b(\frac{x}{y})={\log }_{b}x-{\log }_{b}y\end{array}$  
    ::引号规则: logb(xy) =logbx-logby
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  • Power Rule:  $\begin{array}{r}{\log }_b{x}^n=n{\log }_{b}x\end{array}$  
    ::规则 : logbxn=nlogbx
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  • Inverse:  $\begin{array}{r}{b}^{lo{g}_{b}x}=x\end{array}$  
    ::倒数: 博客bx=x
  
  ::对数属性为: 零指数: logb1=0 身份: logbb=1 产品 规则: logb(xy) = logbx+logby Quotient 规则: logb(xy) = logbx- logby 动力 规则: logbxn=nlogbx 反方向: 博客bx=x
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• To find the inverse of a logarithmic function, you can use the inverse property or convert to exponential form.
  ::要查找对数函数的反向,您可以使用反向属性或转换成指数形式。
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• To find the inverse of an exponential function, you need to take the logarithm of both sides. 
  ::要找到指数函数的逆函数, 您需要使用两侧的对数 。
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• To expand logarithms, write them as a sum or difference of logarithms where the power rule is applied, if necessary. Often, it will be helpful to use the rules in the order quotient rule, product rule, and power rule.
  ::为了扩展对数,必要时在应用权力规则的情况下将其写成对数总和或对数的差数。 通常,使用规则按商序规则、产品规则和权力规则使用会有所帮助。
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• To simplify logarithms, write them as a single logarithm. Often, it will be helpful to use  the rules in the order power rule, product rule, and quotient rule.
  ::为了简化对数, 请把它们写成单对数 。 通常, 使用规则在顺序权力规则、 产品规则和商数规则中使用会有所帮助 。

Solving Exponential and Logarithmic Equations
::解决指数和对数等

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• To solve exponential equations where each side can be expressed with the same base, write each side with the same base and then set the exponents equal to each other.
  ::要解析指数方程式, 每一方方程式可以用相同的基数表示, 以相同的基数写每方方方方程式, 然后设定等值的指数方程式 。
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• To solve exponential equations where each side cannot be expressed with the same base, isolate the term with the variable in the exponent, take any logarithm of both sides of the equation, and then use the power rule of logarithms.
  ::要解析每边无法以相同基数表达的指数方程式, 将术语与引号中的变量分隔开来, 取出方程式两侧的对数, 然后使用对数的功率规则 。
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• To solve logarithmic equations, isolate the term with the logarithm and convert to exponential form. Make sure to check for extraneous solutions. 
  ::要解析对数方程式, 请用对数分离术语, 并转换成指数形式。 请确保检查不相干的解决方案 。

Exponential Growth and Decay Models Including Compound Interest
::指数增长和衰减模型,包括复合利息

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• To find exponential growth or decay models, identify the initial amount and the growth or decay rate.
  ::为了找到指数增长或衰变模型,确定初始数量和增长率或衰变率。
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• You can use either $\begin{array}{r}A(t)=A_0(1+r{)}^t\end{array}$ (growth), $\begin{array}{r}A(t)=A_0(1-r{)}^t\end{array}$ (decay), or $\begin{array}{r}A(t)=A_0{e}^{kt}\end{array}$ (growth or decay depending on $\begin{array}{r}k\end{array}$ ) for exponential models.
  ::您可以使用 A( t) = A0( 1+r) t( 增长) 、 A( t) = A0( 1- r) t( decay) 或 A( t) = A0ekt( 增长或衰减取决于 k) 的指数模型 。
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• If you need to find the amount at a certain time, evaluate the function for that particular time.
  ::如果您在某个时间需要找到金额,请对特定时间的函数进行评价。
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• If you need to find the time until there is a certain amount, solve the exponential equation after substituting the amount into the function.
  ::如果您需要找到时间直到有一定数量,请在将数量替换为函数后解析指数方程式。
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• To calculate compound interest, use the formula $\begin{array}{r}A=P(1+\frac{r}{n}{)}^{nt},\end{array}$  where $\begin{array}{r}A\end{array}$  is the amount after $\begin{array}{r}t\end{array}$  years, $\begin{array}{r}r\end{array}$  is the annual interest rate, and $\begin{array}{r}n\end{array}$  is the number of times you compound per year.
  ::要计算复合利息,请使用公式A=P(1+rn)nt,其中A是年后数额,r是年利率,n是年复数。
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• To calculate interest compounded continuously, use an exponential growth model.
  ::要连续计算利息,请使用指数增长模式。

Looking Back, Looking Forward
::回顾,展望未来

In this chapter, we discussed exponential and logarithmic equations and functions. These functions had many applications, including the amount of money in your bank account, your health based on the pH in your blood, and the sounds made by a piano or band instruments at a concert.
::在本章中,我们讨论了指数和对数方程和函数。这些函数有许多应用,包括银行账户中的钱数、血液中的pH值健康、钢琴或乐队乐器在音乐会上发出的声音。

This chapter concludes our coverage of families of functions. We have looked at linear, quadratic, and higher-degree polynomial functions. We also covered absolute value and piecewise-defined functions. One chapter was dedicated to rational functions, and another to radical functions. Lastly, we considered exponential and logarithmic functions. You now have a collection of functions you can use to model phenomena that happen around you in your day-to-day life.
::本章总结了我们所覆盖的函数序列范围。 我们研究了线性、 二次和更高度的多元函数。 我们还覆盖了绝对值和片断定义的函数。 一章专门论述理性函数,另一章专门论述激进函数。 最后, 我们考虑了指数函数和对数函数。 您现在可以使用一系列函数来模拟日常生活中在你周围发生的现象。

While functions have properties that make them useful for modeling real-world phenomena, we would be limiting ourselves if we considered only  functions. Next we look at a class of relations called "conic sections" that we can use in practical applications as well.   
::虽然功能的属性使得它们有助于模拟现实世界现象,但如果我们只考虑功能,我们就会限制自己。 下一步我们审视一种叫作“ 二次曲线区块”的关系,我们也可以在实际应用中使用。

Chapter Review
::回顾章次审查