剑桥 IGCSE IGCSE, A-Level & AP 练习单预览

指数函数与对数函数

AP 微积分预备 · 第 2 主题

训练
讲义 词汇表
2.1

等差数列与等比数列的变化
大纲
Learning ObjectiveEssential Knowledge

2.1.A
Express arithmetic sequences found in mathematical and contextual scenarios as functions of the whole numbers.

  • 2.1.A.1 A sequence is a function from the whole numbers to the real numbers. Consequently, the graph of a sequence consists of discrete points instead of a curve.
  • 2.1.A.2 Successive terms in an arithmetic sequence have a common difference, or constant rate of change.
  • 2.1.A.3 The general term of an arithmetic sequence with a common difference $d$ is denoted by $a_n$ and is given by $a_n = a_0 + dn$, where $a_0$ is the initial value, or by $a_n = a_k + d(n - k)$, where $a_k$ is the $k$th term of the sequence.

2.1.B
Express geometric sequences found in mathematical and contextual scenarios as functions of the whole numbers.

  • 2.1.B.1 Successive terms in a geometric sequence have a common ratio, or constant proportional change.
  • 2.1.B.2 The general term of a geometric sequence with a common ratio $r$ is denoted by $g_n$ and is given by $g_n = g_0 r^n$, where $g_0$ is the initial value, or by $g_n = g_k r^{(n-k)}$, where $g_k$ is the $k$th term of the sequence.
  • 2.1.B.3 Increasing arithmetic sequences increase equally with each step, whereas increasing geometric sequences increase by a larger amount with each successive step.

来源:美国大学理事会 AP 课程与考试说明

等比级数与收敛

一个数列(sequence)是一个从整数到实数的函数,所以它的图象是离散点,不是一条曲线。

一个等差数列以相等的步攀升;一个等比的乘以一个比
一个等差数列以相等的步攀升;一个等比的乘以一个比
  • 一个等差数列(arithmetic sequence)有一个公差(common difference)$d$(一个恒定的变化率):$a_n = a_0 + dn$,或从一个已知项,$a_n = a_k + d(n-k)$。
  • 一个等比数列(geometric sequence)有一个公比(common ratio)$r$(一个恒定的比例变化):$g_n = g_0\,r^{\,n}$,或 $g_n = g_k\,r^{\,(n-k)}$。

一个递增的等差数列每步增长相同的数额;一个递增的等比数列每步增长一个更大的数额。

Worked example. 一个 $a_0=3$ 和 $d=5$ 的等差数列有 $a_4=3+5(4)=23$。一个 $g_0=2$ 和 $r=3$ 的等比数列有 $g_4=2\cdot3^4=162$ ——加法与乘法使等比的遥遥领先。

探索

Compare arithmetic and geometric growth

An arithmetic sequence adds a fixed step each term (linear); a geometric sequence multiplies by a fixed ratio (exponential). Change the ratio and watch the terms explode or decay.

词汇表 训练
英文 中文 拼音
sequence 数列 shù liè
arithmetic sequence 等差数列 děng chā shù liè
common difference 公差 gōng chāi
geometric sequence 等比数列 děng bǐ shù liè
common ratio 公比 gōng bǐ
练习卷
2.2

线性函数与指数函数的变化
大纲
Learning ObjectiveEssential Knowledge

2.2.A
Construct functions of the real numbers that are comparable to arithmetic and geometric sequences.

  • 2.2.A.1 Linear functions of the form $f(x) = b + mx$ are similar to arithmetic sequences of the form $a_n = a_0 + dn$, as both can be expressed as an initial value ($b$ or $a_0$) plus repeated addition of a constant rate of change, the slope ($m$ or $d$).
  • 2.2.A.2 Similar to arithmetic sequences of the form $a_n = a_k + d(n - k)$, which are based on a known difference, $d$, and a $k$th term, linear functions can be expressed in the form $f(x) = y_i + m(x - x_i)$ based on a known slope, $m$, and a point, $(x_i, y_i)$.
  • 2.2.A.3 Exponential functions of the form $f(x) = ab^x$ are similar to geometric sequences of the form $g_n = g_0 r^n$, as both can be expressed as an initial value ($a$ or $g_0$) times repeated multiplication by a constant proportion ($b$ or $r$).
  • 2.2.A.4 Similar to geometric sequences of the form $g_n = g_k r^{(n-k)}$, which are based on a known ratio, $r$, and a $k$th term, exponential functions can be expressed in the form $f(x) = y_i r^{(x - x_i)}$ based on a known ratio, $r$, and a point, $(x_i, y_i)$.
  • 2.2.A.5 Sequences and their corresponding functions may have different domains.

来源:美国大学理事会 AP 课程与考试说明

数列有连续的表亲:

  • 线性函数(linear functions)$f(x)=b+mx$ 映照等差数列——一个初始值加斜率 $m$ 的重复相加。点形式:$f(x)=y_i+m(x-x_i)$。
  • 指数函数(exponential functions)$f(x)=ab^x$ 映照等比数列——一个初始值乘以底数 $b$ 的重复相乘。点形式:$f(x)=y_i\,r^{\,(x-x_i)}$。

区别:线性 = 重复相加、指数 = 重复相乘。(一个数列和它的函数可能有不同的定义域。)

词汇表 训练
英文 中文 拼音
exponential function 指数函数 zhǐ shù hán shù
练习卷
2.3

指数函数
大纲
Learning ObjectiveEssential Knowledge

2.3.A
Identify key characteristics of exponential functions.

  • 2.3.A.1 The general form of an exponential function is $f(x) = ab^x$, with the initial value $a$, where $a \neq 0$, and the base $b$, where $b > 0$, and $b \neq 1$. When $a > 0$ and $b > 1$, the exponential function is said to demonstrate exponential growth. When $a > 0$ and $0 < b < 1$, the exponential function is said to demonstrate exponential decay.
  • 2.3.A.2 When the natural numbers are input values in an exponential function, the input value specifies the number of factors of the base to be applied to the function's initial value. The domain of an exponential function is all real numbers.
  • 2.3.A.3 Because the output values of exponential functions in general form are proportional over equal-length input-value intervals, exponential functions are always increasing or always decreasing, and their graphs are always concave up or always concave down. Consequently, exponential functions do not have extrema except on a closed interval, and their graphs do not have points of inflection.
  • 2.3.A.4 If the values of the additive transformation function $g(x) = f(x) + k$ of any function $f$ are proportional over equal-length input-value intervals, then $f$ is exponential.

来源:美国大学理事会 AP 课程与考试说明

一般的指数函数(exponential function)是 $f(x)=ab^x$,初始值(initial value)$a\neq 0$ 而底数(base)$b>0,\ b\neq 1$。定义域:所有实数。

指数衰减:每个周期损失一个固定的百分比
指数衰减:每个周期损失一个固定的百分比
  • $a>0,\ b>1$ 给出指数增长(exponential growth);$a>0,\ 0 给出指数衰减(exponential decay)。
  • 输出在等长输入区间上成比例。所以一个指数总是递增或总是递减,而且总是上凹或总是下凹——它没有极值(除了在一个闭区间上)而且没有拐点。
Bacteria under a microscope: exponential models describe populations that grow by a constant factor each time step
Bacteria under a microscope: exponential models describe populations that grow by a constant factor each time step
探索

Explore an exponential curve

y = a·e^(bx) + c

An exponential function changes by a constant factor over equal steps, so it rises (or decays) ever faster. The constant $c$ sets the horizontal asymptote it hugs.

词汇表 训练
英文 中文 拼音
initial value 初始值 chū shǐ zhí
base 底数 dǐ shù
exponential growth 指数增长 zhǐ shù zēng zhǎng
exponential decay 指数衰减 zhǐ shù shuāi jiǎn
练习卷
2.4

指数函数的运算
大纲
Learning ObjectiveEssential Knowledge

2.4.A
Rewrite exponential expressions in equivalent forms.

  • 2.4.A.1 The product property for exponents states that $b^m b^n = b^{(m+n)}$. Graphically, this property implies that every horizontal translation of an exponential function, $f(x) = b^{(x+k)}$, is equivalent to a vertical dilation, $f(x) = b^{(x+k)} = b^x b^k = ab^x$, where $a = b^k$.
  • 2.4.A.2 The power property for exponents states that $\left(b^m\right)^n = b^{(mn)}$. Graphically, this property implies that every horizontal dilation of an exponential function, $f(x) = b^{(cx)}$, is equivalent to a change of the base of an exponential function, $f(x) = \left(b^c\right)^x$, where $b^c$ is a constant and $c \neq 0$.
  • 2.4.A.3 The negative exponent property states that $b^{-n} = \dfrac{1}{b^n}$.
  • 2.4.A.4 The value of an exponential expression involving an exponential unit fraction, such as $b^{(1/k)}$ where $k$ is a natural number, is the $k$th root of $b$, when it exists.

来源:美国大学理事会 AP 课程与考试说明

指数规则重塑指数表达式并连接到图象变换:

  • 积: $b^m b^n = b^{m+n}$。一个水平移动 $b^{x+k}$ 等于一个竖直拉伸 $ab^x$,$a=b^k$。
  • 幂: $(b^m)^n = b^{mn}$。一个水平拉伸 $b^{cx}$ 等于一个底数改变 $(b^c)^x$。
  • 负指数: $b^{-n}=\dfrac{1}{b^n}$。
  • 单位分数指数: $b^{1/k}=\sqrt[k]{b}$(第 $k$ 个方根(root))。
词汇表 训练
英文 中文 拼音
root 方根 fāng gēn
练习卷
2.5

指数函数的情境与数据建模
大纲
Learning ObjectiveEssential Knowledge

2.5.A
Construct a model for situations involving proportional output values over equal-length input-value intervals.

  • 2.5.A.1 Exponential functions model growth patterns where successive output values over equal-length input-value intervals are proportional. When the input values are whole numbers, exponential functions model situations of repeated multiplication of a constant to an initial value.
  • 2.5.A.2 A constant may need to be added to the dependent variable values of a data set to reveal a proportional growth pattern.
  • 2.5.A.3 An exponential function model can be constructed from an appropriate ratio and initial value or from two input-output pairs. The initial value and the base can be found by solving a system of equations resulting from the two input-output pairs.
  • 2.5.A.4 Exponential function models can be constructed by applying transformations to $f(x) = ab^x$ based on characteristics of a contextual scenario or data set.
  • 2.5.A.5 Exponential function models can be constructed for a data set with technology using exponential regressions.
  • 2.5.A.6 The natural base $e$, which is approximately $2.718$, is often used as the base in exponential functions that model contextual scenarios.

来源:美国大学理事会 AP 课程与考试说明

指数为在等间隔上以一个恒定比例增长(重复相乘)的量建模。

  • 从一个比和一个初始值,或从两个点(为 $a$ 和 $b$ 解这个系统)构建一个模型。
  • 有时一个常数必须被加到数据以揭示比例模式。
  • 在技术上用指数回归(exponential regression)以拟合一个数据集。
  • 自然底数(natural base)$e\approx 2.718$ 是现实世界模型的标准底数。
词汇表 训练
英文 中文 拼音
exponential regression 指数回归 zhǐ shù huí guī
natural base 自然底数 zì rán dǐ shù
练习卷
2.6

竞争函数模型的验证
大纲
Learning ObjectiveEssential Knowledge

2.6.A
Construct linear, quadratic, and exponential models based on a data set.

  • 2.6.A.1 Two variables in a data set that demonstrate a slightly changing rate of change can be modeled by linear, quadratic, and exponential function models.
  • 2.6.A.2 Models can be compared based on contextual clues and applicability to determine which model is most appropriate.

2.6.B
Validate a model constructed from a data set.

  • 2.6.B.1 A model is justified as appropriate for a data set if the graph of the residuals of a regression, the residual plot, appear without pattern.
  • 2.6.B.2 The difference between the predicted and actual values is the error in the model. Depending on the data set and context, it may be more appropriate to have an underestimate or overestimate for any given interval.

来源:美国大学理事会 AP 课程与考试说明

当一个变化率只略微转变时,线性、二次和指数模型可能都看似拟合。用上下文和拟合质量选择:

  • 一个模型是合适的,若它的残差图(residual plot)显示没有模式(一个残差(residual)是实际减预测)。
  • 误差是预测和实际之间的差距;上下文决定一个高估还是低估更安全。
词汇表 训练
英文 中文 拼音
residual plot 残差图 cán chà tú
residual 残差 cán chà
练习卷
2.7

复合函数
大纲
Learning ObjectiveEssential Knowledge

2.7.A
Evaluate the composition of two or more functions for given values.

  • 2.7.A.1 If $f$ and $g$ are functions, the composite function $f \circ g$ maps a set of input values to a set of output values such that the output values of $g$ are used as input values of $f$. For this reason, the domain of the composite function is restricted to those input values of $g$ for which the corresponding output value is in the domain of $f$. $(f \circ g)(x)$ can also be represented as $f(g(x))$.
  • 2.7.A.2 Values for the composite function $f \circ g$ can be calculated or estimated from the graphical, numerical, analytical, or verbal representations of $f$ and $g$ by using output values from $g$ as input values for $f$.
  • 2.7.A.3 The composition of functions is not commutative; that is, $f \circ g$ and $g \circ f$ are typically different functions; therefore, $f(g(x))$ and $g(f(x))$ are typically different values.
  • 2.7.A.4 If the function $f(x) = x$ is composed with any function $g$, the resulting composite function is the same as $g$; that is, $g(f(x)) = f(g(x)) = g(x)$. The function $f(x) = x$ is called the identity function. When composing two functions, the identify function acts in the same way as $0$, the additive identity, when adding two numbers and $1$, the multiplicative identity, when multiplying two numbers.

2.7.B
Construct a representation of the composition of two or more functions.

  • 2.7.B.1 Function composition is useful for relating two quantities that are not directly related by an existing formula.
  • 2.7.B.2 When analytic representations of the functions $f$ and $g$ are available, an analytic representation of $f(g(x))$ can be constructed by substituting $g(x)$ for every instance of $x$ in $f$.
  • 2.7.B.3 A numerical or graphical representation of $f \circ g$ can often be constructed by calculating or estimating values for $(x, f(g(x)))$.

2.7.C
Rewrite a given function as a composition of two or more functions.

  • 2.7.C.1 Functions given analytically can often be decomposed into less complicated functions. When properly decomposed, the variable in one function should replace each instance of the function with which it was composed.
  • 2.7.C.2 An additive transformation of a function, $f$, that results in vertical and horizontal translations can be understood as the composition of $g(x) = x + k$ with $f$.
  • 2.7.C.3 A multiplicative transformation of a function, $f$, that results in vertical and horizontal dilations can be understood as the composition of $g(x) = kx$ with $f$.

来源:美国大学理事会 AP 课程与考试说明

复合函数(composite function)$(f\circ g)(x)=f(g(x))$ 把 $g$ 的输出喂进 $f$。它的定义域是 $g$ 的、输出位于 $f$ 定义域里的那些输入。

一个函数作为一台机器;它的反函数向后运行这台机器
一个函数作为一台机器;它的反函数向后运行这台机器
  • 复合不可交换:$f(g(x))$ 和 $g(f(x))$ 通常不同。
  • 要解析地构建 $f(g(x))$,把 $f$ 里的每个 $x$ 代入 $g(x)$。
  • 恒等函数(identity function)$f(x)=x$ 在复合下使任何函数不变。
  • 一个函数也能被分解成更简单的片段——对把一个加性移动看作与 $x+k$ 复合、或一个伸缩看作与 $kx$ 复合有用。
Matryoshka dolls: function composition nests layers — evaluate the inside first
Matryoshka dolls: function composition nests layers — evaluate the inside first
词汇表 训练
英文 中文 拼音
composite function 复合函数 fù hé hán shù
identity function 恒等函数 héng děng hán shù
练习卷
2.8

反函数
大纲
Learning ObjectiveEssential Knowledge

2.8.A
Determine the input-output pairs of the inverse of a function.

  • 2.8.A.1 On a specified domain, a function, $f$, has an inverse function, or is invertible, if each output value of $f$ is mapped from a unique input value. The domain of a function may be restricted in many ways to make the function invertible.
  • 2.8.A.2 An inverse function can be thought of as a reverse mapping of the function. An inverse function, $f^{-1}$, maps the output values of a function, $f$, on its invertible domain to their corresponding input values; that is, if $f(a) = b$, then $f^{-1}(b) = a$. Alternately, on its invertible domain, if a function consists of input-output pairs $(a, b)$, then the inverse function consists of input-output pairs $(b, a)$.

2.8.B
Determine the inverse of a function on an invertible domain.

  • 2.8.B.1 The composition of a function, $f$, and its inverse function, $f^{-1}$, is the identity function; that is, $f\left(f^{-1}(x)\right) = f^{-1}(f(x)) = x$.
  • 2.8.B.2 On a function's invertible domain, the function's range and domain are the inverse function's domain and range, respectively. The inverse of the table of values of $y = f(x)$ can be found by reversing the input-output pairs; that is, $(a, b)$ corresponds to $(b, a)$.
  • 2.8.B.3 The inverse of the graph of the function $y = f(x)$ can be found by reversing the roles of the $x$- and $y$-axes; that is, by reflecting the graph of the function over the graph of the identity function $h(x) = x$.
  • 2.8.B.4 The inverse of the function can be found by determining the inverse operations to reverse the mapping. One method for finding the inverse of the function $f$ is reversing the roles of $x$ and $y$ in the equation $y = f(x)$, then solving for $y = f^{-1}(x)$.
  • 2.8.B.5 In addition to limiting the domain of a function to obtain an inverse function, contextual restrictions may also limit the applicability of an inverse function.

来源:美国大学理事会 AP 课程与考试说明

反函数是关于 y=x 的反射
指数与对数互为反函数

一个函数在每个输出来自一个唯一输入的定义域上是可逆(invertible)的(你可以限制定义域来强制这个)。反函数(inverse function)$f^{-1}$ 反转这个映射:若 $f(a)=b$ 那么 $f^{-1}(b)=a$。

  • $f\big(f^{-1}(x)\big)=f^{-1}\big(f(x)\big)=x$(复合给出恒等)。
  • 定义域和值域交换:在一张表里把每个 $(a,b)$ 反转成 $(b,a)$;把图象在线 $y=x$ 上反射。
  • 要求一个公式:在 $y=f(x)$ 里交换 $x$ 和 $y$,然后解 $y$。上下文可能进一步限制反函数在哪里适用。
词汇表 训练
英文 中文 拼音
invertible 可逆 kě nì
inverse function 反函数 fǎn hán shù
练习卷
2.9

对数表达式
大纲
Learning ObjectiveEssential Knowledge

2.9.A
Evaluate logarithmic expressions.

  • 2.9.A.1 The logarithmic expression $\log_b c$ is equal to, or represents, the value that the base $b$ must be exponentially raised to in order to obtain the value $c$. That is, $\log_b c = a$ if and only if $b^a = c$, where $a$ and $c$ are constants, $b > 0$, and $b \neq 1$. (when the base of a logarithmic expression is not specified, it is understood as the common logarithm with a base of $10$)
  • 2.9.A.2 The values of some logarithmic expressions are readily accessible through basic arithmetic while other values can be estimated through the use of technology.
  • 2.9.A.3 On a logarithmic scale, each unit represents a multiplicative change of the base of the logarithm. For example, on a standard scale, the units might be $0, \; 1, \; 2, \; \ldots$, while on a logarithmic scale, using logarithm base $10$, the units might be $10^0, \; 10^1, \; 10^2, \; \ldots$.

来源:美国大学理事会 AP 课程与考试说明

对数(logarithm)回答"什么指数?":$\log_b c = a$ 恰好意味着 $b^a = c$(以 $b>0,\ b\neq 1$)。一个未写的底数意味着常用对数(common logarithm)(底数 $10$)。在一个对数尺度上,每个单位是底数的一个乘性步(……,$10^0,10^1,10^2,$ ……)。

词汇表 训练
英文 中文 拼音
logarithm 对数 duì shù
common logarithm 常用对数 cháng yòng duì shù
练习卷
2.10

指数函数的反函数
大纲
Learning ObjectiveEssential Knowledge

2.10.A
Construct representations of the inverse of an exponential function with an initial value of 1.

  • 2.10.A.1 The general form of a logarithmic function is $f(x) = a\log_b x$, with base $b$, where $b > 0$, $b \neq 1$, and $a \neq 0$.
  • 2.10.A.2 The way in which input and output values vary together have an inverse relationship in exponential and logarithmic functions. Output values of general-form exponential functions change proportionately as input values increase in equal-length intervals. However, input values of general-form logarithmic functions change proportionately as output values increase in equal-length intervals. Alternately, exponential growth is characterized by output values changing multiplicatively as input values change additively, whereas logarithmic growth is characterized by output values changing additively as input values change multiplicatively.
  • 2.10.A.3 $f(x) = \log_b x$ and $g(x) = b^x$, where $b > 0$ and $b \neq 1$, are inverse functions. That is, $g(f(x)) = f(g(x)) = x$.
  • 2.10.A.4 The graph of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$, is a reflection of the graph of the exponential function $g(x) = b^x$, where $b > 0$ and $b \neq 1$, over the graph of the identity function $h(x) = x$.
  • 2.10.A.5 If $(s, \; t)$ is an ordered pair of the exponential function $g(x) = b^x$, where $b > 0$ and $b \neq 1$, then $(t, \; s)$ is an ordered pair of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$.

来源:美国大学理事会 AP 课程与考试说明

对数是指数的反函数:$y=b^x$ 和 $y=\log_b x$ 互相撤销,所以它们的图象是在 $y=x$ 上的反射。因此 $\log_b(b^x)=x$ 而 $b^{\log_b x}=x$。自然对数(natural logarithm)$\ln x = \log_e x$ 是 $e^x$ 的反函数。

e^x 和 ln x 是彼此在线 y = x 里的反射
e^x 和 ln x 是彼此在线 y = x 里的反射
词汇表 训练
英文 中文 拼音
natural logarithm 自然对数 zì rán duì shù
练习卷
2.11

对数函数
大纲
Learning ObjectiveEssential Knowledge

2.11.A
Identify key characteristics of logarithmic functions.

  • 2.11.A.1 The domain of a logarithmic function in general form is any real number greater than zero, and its range is all real numbers.
  • 2.11.A.2 Because logarithmic functions are inverses of exponential functions, logarithmic functions are also always increasing or always decreasing, and their graphs are either always concave up or always concave down. Consequently, logarithmic functions do not have extrema except on a closed interval, and their graphs do not have points of inflection.
  • 2.11.A.3 The additive transformation function $g(x) = f(x + k)$, where $k \neq 0$, of a logarithmic function $f$ in general form does not have input values that are proportional over equal-length output-value intervals. However, if the input values of the additive transformation function, $g(x) = f(x + k)$, of any function $f$ are proportional over equal-length output value intervals, then $f$ is logarithmic.
  • 2.11.A.4 With their limited domain, logarithmic functions in general form are vertically asymptotic to $x = 0$, with an end behavior that is unbounded. That is, for a logarithmic function in general form, $\lim\limits_{x \to 0^+} a\log_b x = \pm\infty$ and $\lim\limits_{x \to \infty} a\log_b x = \pm\infty$.

来源:美国大学理事会 AP 课程与考试说明

对数函数(logarithmic function)$f(x)=\log_b x$ 有定义域 $x>0$ 而值域是所有实数。它总是递增(对 $b>1$)或总是递减(对 $0)、总是朝一个方向凹,在 $x=0$ 有一条垂直渐近线——指数的水平渐近线的镜像。它对大的 $x$ 增长得很慢。

探索

Explore a logarithmic curve

y = a·ln(x − b) + c

A logarithm is the inverse of an exponential: it grows without bound but ever slower, with a vertical asymptote where its input hits zero.

词汇表 训练
英文 中文 拼音
logarithmic function 对数函数 duì shù hán shù
练习卷
2.12

对数函数的运算
大纲
Learning ObjectiveEssential Knowledge

2.12.A
Rewrite logarithmic expressions in equivalent forms.

  • 2.12.A.1 The product property for logarithms states that $\log_b(xy) = \log_b x + \log_b y$. Graphically, this property implies that every horizontal dilation of a logarithmic function, $f(x) = \log_b(kx)$, is equivalent to a vertical translation, $f(x) = \log_b(kx) = \log_b k + \log_b x = a + \log_b x$, where $a = \log_b k$.
  • 2.12.A.2 The power property for logarithms states that $\log_b x^n = n\log_b x$. Graphically, this property implies that raising the input of a logarithmic function to a power, $f(x) = \log_b x^k$, results in a vertical dilation, $f(x) = \log_b x^k = k\log_b x$.
  • 2.12.A.3 The change of base property for logarithms states that $\log_b x = \dfrac{\log_a x}{\log_a b}$, where $a > 0$ and $a \neq 1$. This implies that all logarithmic functions are vertical dilations of each other.
  • 2.12.A.4 The function $f(x) = \ln x$ is a logarithmic function with the natural base $e$; that is, $\ln x = \log_e x$.

来源:美国大学理事会 AP 课程与考试说明

对数性质反转指数规则:

$$\log_b(xy)=\log_b x+\log_b y,\quad \log_b\!\frac{x}{y}=\log_b x-\log_b y,\quad \log_b(x^n)=n\log_b x.$$
换底(change-of-base)公式让你能用技术计算任何对数:$\log_b x = \dfrac{\log x}{\log b}=\dfrac{\ln x}{\ln b}$。

词汇表 训练
英文 中文 拼音
change-of-base 换底 huàn dǐ
练习卷
2.13

指数与对数方程和不等式
大纲
Learning ObjectiveEssential Knowledge

2.13.A
Solve exponential and logarithmic equations and inequalities.

  • 2.13.A.1 Properties of exponents, properties of logarithms, and the inverse relationship between exponential and logarithmic functions can be used to solve equations and inequalities involving exponents and logarithms.
  • 2.13.A.2 When solving exponential and logarithmic equations found through analytical or graphical methods, the results should be examined for extraneous solutions precluded by the mathematical or contextual limitations.
  • 2.13.A.3 Logarithms can be used to rewrite expressions involving exponential functions in different ways that may reveal helpful information. Specifically, $b^x = c^{(\log_c b)(x)}$.

2.13.B
Construct the inverse function for exponential and logarithmic functions.

  • 2.13.B.1 The function $f(x) = ab^{(x-h)} + k$ is a combination of additive transformations of an exponential function in general form. The inverse of $y = f(x)$ can be found by determining the inverse operations to reverse the mapping.
  • 2.13.B.2 The function $f(x) = a\log_b(x - h) + k$ is a combination of additive transformations of a logarithmic function in general form. The inverse of $y = f(x)$ can be found by determining the inverse operations to reverse the mapping.

来源:美国大学理事会 AP 课程与考试说明

要解,用指数和对数互为反函数的事实:

  • 隔离指数,然后对两侧取一个对数(用幂性质把指数拿下来)。
  • 隔离对数,然后对两侧取指数。
  • 总是检查增根(extraneous solutions)——一个对数的自变量必须保持正。

Worked example. 解 $2\cdot3^x=54$。除以 $2$:$3^x=27=3^3$,所以 $x=3$。当两侧不是整齐的幂时,改取对数:$5^x=20$ 给出 $x=\dfrac{\ln 20}{\ln 5}\approx1.86$。

练习卷
2.14

对数函数的情境与数据建模
大纲
Learning ObjectiveEssential Knowledge

2.14.A
Construct a logarithmic function model.

  • 2.14.A.1 Logarithmic functions are inverses of exponential functions and can be used to model situations involving proportional growth, or repeated multiplication, where the input values change proportionally over equal-length output-value intervals. Alternately, if the output value is a whole number, it indicates how many times the initial value has been multiplied by the proportion.
  • 2.14.A.2 A logarithmic function model can be constructed from an appropriate proportion and a real zero or from two input-output pairs.
  • 2.14.A.3 Logarithmic function models can be constructed by applying transformations to $f(x) = a\log_b x$ based on characteristics of a context or data set.
  • 2.14.A.4 Logarithmic function models can be constructed for a data set with technology using logarithmic regressions.
  • 2.14.A.5 The natural logarithm function is often useful in modeling real-world phenomena.
  • 2.14.A.6 Logarithmic function models can be used to predict values for the dependent variable.

来源:美国大学理事会 AP 课程与考试说明

对数为在巨大的乘性范围上变化的量(声音、酸强度、地震)建模。从数据构建一个对数模型,并对一个数据集用对数回归(logarithmic regression)。因为一个对数压缩大值,它把比例增长变成一个直线模式——半对数图背后的思想。

练习卷
2.15

半对数图
大纲
Learning ObjectiveEssential Knowledge

2.15.A
Determine if an exponential model is appropriate by examining a semi-log plot of a data set.

  • 2.15.A.1 In a semi-log plot, one of the axes is logarithmically scaled. When the $y$-axis of a semi-log plot is logarithmically scaled, data or functions that demonstrate exponential characteristics will appear linear.
  • 2.15.A.2 An advantage of semi-log plots is that a constant never needs to be added to the dependent variable values to reveal that an exponential model is appropriate.

2.15.B
Construct the linearization of exponential data.

  • 2.15.B.1 Techniques used to model linear functions can be applied to a semi-log graph.
  • 2.15.B.2 For an exponential model of the form $y = ab^x$, the corresponding linear model for the semi-log plot is $y = (\log_n b)x + \log_n a$, where $n > 0$ and $n \neq 1$. Specifically, the linear rate of change is $\log_n b$, and the initial linear value is $\log_n a$.

来源:美国大学理事会 AP 课程与考试说明

一个半对数图(semi-log plot)把输出放在一个对数轴上而输入放在一个正常轴上。在这些轴上,一个指数函数 $y=ab^x$ 变成一条直线,因为 $\log y = \log a + (\log b)\,x$ 在 $x$ 里是线性的。所以:若数据在一个半对数图上看起来线性,一个指数模型拟合;线的斜率给出 $\log b$ 而它的截距给出 $\log a$。

词汇表 训练
英文 中文 拼音
semi-log plot 半对数图 bàn duì shù tú
练习卷
2.15

考试技巧
  • 把等差(加一个公差)与等比(乘一个公比)数列及它们的线性/指数函数表亲区分开。
  • 指数 = 重复相乘,所以它最终超过任何线性或多项式模型。
  • 对数是指数的反函数($\log_b c=a\Leftrightarrow b^a=c$);用它来解 $b^x=k$。
  • 应用对数律(积、商、幂)和换底公式;一个对数的自变量必须是正的。
  • 在一个半对数图上一个指数模型变成一条直线。

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