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Practice练习题

Exponential and Logarithmic Functions指数函数与对数函数

Practice Questions · SAT · AP-Feeder · ON / BC Provincial Styles练习题集 · SAT 风格 · AP 衔接 · 安大略 / 不列颠哥伦比亚省考风格

EASY易 MEDIUM中 HARD难 🇺🇸 US美 🇨🇦 ON安 🇨🇦 BC卑 SAT-style MCQSAT 风格选择题 AP-feeder FRQAP 衔接简答题 ON Provincial-style安大略省考风格 BC Provincial-style卑诗省考风格 Honors荣誉级
Name:姓名：Date:日期：
PART I  ·  SHORT RESPONSE第一部分  ·  短答题SAT-style MCQ + ON/BC short answer · 22 marksSAT 风格选择题 + 安/卑省考短答 · 共 22 分

Section A · Short ResponseA 部分 · 短答题

Mix of multiple-choice and short-answer items. For MCQs, circle the letter; show enough work in the margin that a marker could verify. State exact forms where possible (e.g. $\log_{3} 16$, $\ln 5 / \ln 2$) before any decimal rounding. No calculator on Q1-Q3; calculator permitted on Q4-Q5.本节包含选择题与短答题。选择题请圈出字母答案，并在空白处写下足以让阅卷人核对的过程。在小数舍入前尽量给出精确形式（如 $\log_{3} 16$、$\ln 5 / \ln 2$）。Q1–Q3 不可使用计算器；Q4–Q5 可用计算器。

Q1EASY易 🇺🇸 US美 SAT-style MCQSAT 风格选择题 §1 Exp Parent & Asymptote指数母函数与渐近线 · HSF-IF.C.7e [3 marks][3 分]

The graph of $f(x) = 4 \cdot 2^{x}$ is sketched in the $xy$-plane. Which statement correctly describes its $y$-intercept and horizontal asymptote?在 $xy$ 平面内画出指数函数 $f(x) = 4 \cdot 2^{x}$ 的图像。下列哪一项正确描述其 $y$ 轴截距与水平渐近线？

  1. (A) $y$-intercept $(0, 1)$; asymptote $y = 0$$y$ 轴截距 $(0, 1)$；渐近线 $y = 0$
  2. (B) $y$-intercept $(0, 4)$; asymptote $y = 0$$y$ 轴截距 $(0, 4)$；渐近线 $y = 0$
  3. (C) $y$-intercept $(0, 4)$; asymptote $y = 4$$y$ 轴截距 $(0, 4)$；渐近线 $y = 4$
  4. (D) $y$-intercept $(0, 2)$; asymptote $x = 0$$y$ 轴截距 $(0, 2)$；渐近线 $x = 0$
Q2EASY易 🇺🇸 US美 SAT-style MCQSAT 风格选择题 §2 Same-Base Exp Eq同底指数方程 · HSF-LE.A.4 [3 marks][3 分]

Solve $9^{x + 1} = 27$ exactly, without a calculator.不使用计算器，精确求解 $9^{x + 1} = 27$。

  1. (A) $x = 1/2$
  2. (B) $x = 2$
  3. (C) $x = 3/2$
  4. (D) $x = -1/2$
Q3EASY易 🇺🇸 US美 SAT-style MCQSAT 风格选择题 §3 Log as Inverse对数即反函数 · HSF-BF.B.5 (+) [3 marks][3 分]

Which equation is equivalent to $\log_{5}(x) = 3$?下列哪一项方程等价于 $\log_{5}(x) = 3$？

  1. (A) $3^{5} = x$
  2. (B) $5^{3} = x$
  3. (C) $5 \cdot 3 = x$
  4. (D) $x^{3} = 5$
Q4MEDIUM中 🇨🇦 ON安 ON Provincial-style安大略省考风格 §4 Laws of Logs对数运算律 · MHF4U Exp & Log指数与对数 [6 marks][6 分]

Use the laws of logarithms to evaluate or simplify each expression. State which law(s) you applied at each step.利用对数运算律对下列各式求值或化简，并在每一步注明所用的运算律。

(a) Evaluate exactly: $\log_{2}(32) - \log_{2}(4)$.精确求值：$\log_{2}(32) - \log_{2}(4)$。 [2]
(b) Condense to a single logarithm: $2 \log_{3}(x) + \log_{3}(5) - \log_{3}(y)$.合并为单个对数：$2 \log_{3}(x) + \log_{3}(5) - \log_{3}(y)$。 [2]
(c) Expand and simplify: $\log_{10}\!\left(\dfrac{100 \, x^{3}}{\sqrt{y}}\right)$.展开并化简：$\log_{10}\!\left(\dfrac{100 \, x^{3}}{\sqrt{y}}\right)$。 [2]
Q5MEDIUM中 🇨🇦 BC卑 BC Provincial-style卑诗省考风格 §5 Different-Base Exp异底指数方程 · BC PC12 exp eqsBC PC12 指数方程 [7 marks][7 分]

Solve each exponential equation. Give exact answers in the form $\log_{b}(\,\cdot\,)$ or $\ln(\,\cdot\,) / \ln(\,\cdot\,)$, then a decimal to three places.求解下列指数方程。请先给出 $\log_{b}(\,\cdot\,)$ 或 $\ln(\,\cdot\,) / \ln(\,\cdot\,)$ 形式的精确解，再写出保留三位小数的近似值。

(a) $3^{x} = 20$. [3]
(b) $5 \cdot 2^{x} = 90$. [2]
(c) $7^{x - 1} = 4^{x}$. State the change-of-base step explicitly.$7^{x - 1} = 4^{x}$。请明确写出换底公式的步骤。 [2]
PART II  ·  EXTENDED RESPONSE第二部分  ·  简答题AP-feeder FRQ + honors · 36 marksAP 衔接简答题 + 荣誉级 · 共 36 分

Section B · Extended ResponseB 部分 · 简答题

Show every algebraic step. For log equations, check every candidate against the domain of each logarithm and discard extraneous solutions. State whether you used same-base or take-the-log on every exponential equation. No calculator on Q6 and Q8; calculator permitted on Q7 and Q9.每一步代数运算都要写出。对对数方程，每个候选解都要对照每个对数的定义域进行验证，并舍弃增根。对每个指数方程都要注明所用方法（同底法或两边取对数）。Q6 与 Q8 不可使用计算器；Q7 与 Q9 可用计算器。

Q6MEDIUM中 🇺🇸 US美 AP-feeder FRQAP 衔接简答题 §4 Change-of-Base换底公式 · HSF-BF.B.5 (+) [8 marks][8 分]

Let $L = \log_{4}(50)$.设 $L = \log_{4}(50)$。

(a) Write $L$ in change-of-base form using the natural logarithm, then approximate $L$ to four decimal places.用自然对数把 $L$ 写成换底公式形式，再近似到四位小数。 [3]
(b) Without using a calculator, prove the identity $\log_{b}(x^{p}) = p \log_{b}(x)$ for $b > 0$, $b \ne 1$, $x > 0$ by starting from $y = \log_{b}(x^{p}) \iff b^{y} = x^{p}$ and applying laws of exponents.不用计算器，从 $y = \log_{b}(x^{p}) \iff b^{y} = x^{p}$ 出发，利用指数律证明恒等式 $\log_{b}(x^{p}) = p \log_{b}(x)$（其中 $b > 0$，$b \ne 1$，$x > 0$）。 [3]
(c) Use part (b) to evaluate exactly $\log_{4}(50) - \log_{4}(2) - \log_{4}(25)$ without a calculator.利用 (b) 的结论，不用计算器精确求出 $\log_{4}(50) - \log_{4}(2) - \log_{4}(25)$。 [2]
Q7HARD难 🇨🇦 ON安 ON Provincial-style安大略省考风格 §5 Log Equations + Extraneous对数方程 + 增根 · MHF4U Exp & Log指数与对数 [9 marks][9 分]

Solve each logarithmic equation algebraically. For each candidate solution, verify the argument of every logarithm is strictly positive; discard extraneous solutions and state the final solution set.用代数方法求解下列对数方程。对每个候选解，需验证每个对数的真数严格大于零；舍弃增根并写出最终解集。

(a) $\log_{2}(x) + \log_{2}(x - 2) = 3$. [4]
(b) $\log_{5}(2 x + 3) - \log_{5}(x - 1) = 1$. [3]
(c) State, in one sentence, why checking the domain of each logarithm is required and not optional.用一句话说明为什么验证每个对数的定义域是必需的，而不是可选的。 [2]
Q8HARD难Honors荣誉级 🇺🇸 US美 🇨🇦 BC卑 AP-feeder FRQAP 衔接简答题 §6 Base $e$ & Natural Log底数 $e$ 与自然对数 · BC PC12 / HSF-LE.A.4 [10 marks][10 分]

Let $g(x) = 3 e^{2 x} - 5$.设 $g(x) = 3 e^{2 x} - 5$。

(a) State the horizontal asymptote of $y = g(x)$ and the $y$-intercept.写出 $y = g(x)$ 的水平渐近线与 $y$ 轴截距。 [2]
(b) Solve $g(x) = 16$ exactly using the natural logarithm, then approximate to three decimal places.用自然对数精确求解 $g(x) = 16$，再保留三位小数。 [3]
(c) Determine the inverse $g^{-1}(x)$ explicitly in terms of $\ln$. State the domain and range of $g^{-1}$.用 $\ln$ 显式求出反函数 $g^{-1}(x)$，并写出 $g^{-1}$ 的定义域与值域。 [3]
(d) Cite the BC PC12 Big Idea on inverses (one sentence) and explain why solving $g(x) = 16$ in (b) is precisely an instance of it.引用 BC PC12 关于反函数的大概念（一句话），并解释 (b) 中求解 $g(x) = 16$ 为何正是该概念的具体实例。 [2]
Q9HARD难Honors荣誉级 🇺🇸 US美 🇨🇦 BC卑 AP-feeder FRQAP 衔接简答题 §6 Continuous Compounding连续复利 · BC PC12 base $e$BC PC12 底数 $e$ / HSF-LE.B.5 [9 marks][9 分]

A bank advertises a nominal annual interest rate of $4.5\%$ on a savings account with continuous compounding. The balance after $t$ years on an initial principal $P_{0}$ is modeled by $P(t) = P_{0} \, e^{0.045 \, t}$.某银行的储蓄账户标称年利率为 $4.5\%$，按连续复利计息。本金 $P_{0}$ 经过 $t$ 年后的余额由 $P(t) = P_{0} \, e^{0.045 \, t}$ 给出。

(a) A customer deposits US$$2{,}000$. Compute the balance after $10$ years to the nearest cent.某客户存入 US$$2{,}000$。计算 $10$ 年后的余额，精确到分。 [2]
(b) Solve symbolically for the doubling time $T$ such that $P(T) = 2 P_{0}$. Express $T$ exactly using $\ln$, then approximate to two decimal places.符号求解倍增时间 $T$，使 $P(T) = 2 P_{0}$。用 $\ln$ 给出精确表达式，再保留两位小数。 [3]
(c) Interpret the parameters $P_{0}$ and the rate $0.045$ in the context of the situation. Cite HSF-LE.B.5: "interpret the parameters in… an exponential function in terms of a context."结合情境解释参数 $P_{0}$ 与利率 $0.045$ 的实际含义。援引 HSF-LE.B.5："结合情境解释指数函数中的各参数。" [2]
(d) A rival bank offers $4.5\%$ compounded annually instead of continuously. After $10$ years on the same US$$2{,}000$ principal, which account holds more, and by how much (nearest cent)?另一家银行同样提供 $4.5\%$ 的年利率，但按年复利而非连续复利。同样 US$$2{,}000$ 本金，$10$ 年后哪个账户余额更多？多多少（精确到分）？ [2]
PART III  ·  MODELING / APPLIED第三部分  ·  建模与应用Universal · 32 marks通用题型 · 共 32 分

Section C · Modeling and ApplicationsC 部分 · 建模与应用

Name your variables (with units) before writing equations. State whether the situation is growth or decay and identify the growth/decay factor before substituting. Conclude with a one-sentence answer in context, rounded sensibly. Calculator permitted throughout Part III.在写方程前，先定义变量名（含单位）。代入数值前先注明情境为增长还是衰减，并写出增长 / 衰减倍数。以一句结合情境的话作答，合理舍入。第三部分全程可用计算器。

Q10MEDIUM中 🇺🇸 US美 AP-feeder FRQAP 衔接简答题 §7 Population Growth人口增长 · HSF-LE.A.1c / HSF-LE.A.2 [10 marks][10 分]

A coastal town's population grows by a constant percent rate of $2.4\%$ per year. In the year $2020$ its population was $48{,}000$. Let $P(t)$ denote the population $t$ years after $2020$.某海滨城镇的人口以每年 $2.4\%$ 的恒定百分率增长。$2020$ 年其人口为 $48{,}000$。设 $P(t)$ 表示 $2020$ 年后 $t$ 年的人口数。

(a) Write the explicit model $P(t) = a \cdot b^{t}$. State the values of $a$ and $b$ and identify which represents the initial value and which the annual growth factor.写出显式模型 $P(t) = a \cdot b^{t}$。给出 $a$ 与 $b$ 的值，并说明哪个代表初始值、哪个代表年增长倍数。 [3]
(b) Predict the population in $2030$ to the nearest whole person.预测 $2030$ 年的人口数，精确到整人。 [2]
(c) Determine, using logarithms, the year in which the population is first projected to exceed $75{,}000$. Round your value of $t$ up to the next whole year.用对数确定人口首次超过 $75{,}000$ 的年份。$t$ 值向上取整到下一整年。 [3]
(d) Cite HSF-LE.A.1c to explain why "$2.4\%$ per year" produces an exponential rather than a linear model.援引 HSF-LE.A.1c，说明为何"每年 $2.4\%$"产生的是指数模型而非线性模型。 [2]
Q11MEDIUM中 🇨🇦 ON安 ON Provincial-style安大略省考风格 §7 Radioactive Half-Life放射性半衰期 · MHF4U Exp & Log指数与对数 [11 marks][11 分]

Iodine-131 has a half-life of approximately $8.02$ days. A medical sample initially contains $40$ mg of I-131. Let $M(t)$ be the mass remaining (in mg) after $t$ days.碘-131 的半衰期约为 $8.02$ 天。某医用样本初始含 $40$ mg 的 I-131。设 $M(t)$ 为 $t$ 天后剩余的质量（毫克）。

(a) Write $M(t)$ in the half-life form $M(t) = M_{0} \cdot \left(\tfrac{1}{2}\right)^{t / h}$. State $M_{0}$ and $h$.把 $M(t)$ 写成半衰期形式 $M(t) = M_{0} \cdot \left(\tfrac{1}{2}\right)^{t / h}$，并写出 $M_{0}$ 与 $h$。 [2]
(b) Compute the mass remaining after $30$ days, to two decimal places.计算 $30$ 天后的剩余质量，保留两位小数。 [2]
(c) Determine, using logarithms, the time required for the sample to decay to $5$ mg. Round to the nearest tenth of a day.用对数确定样本衰减到 $5$ mg 所需的时间，精确到 $0.1$ 天。 [4]
(d) State the practical (situational) domain of $M(t)$ in interval notation, and identify whether this is exponential growth or decay.用区间记号写出 $M(t)$ 的实际（情境）定义域，并判断这是指数增长还是指数衰减。 [2]
(e) State one assumption baked into the half-life model that is in practice only approximately true.写出半衰期模型中蕴含的一个在实际中只近似成立的假设。 [1]
Q12HARD难 🇨🇦 BC卑 🇺🇸 US美 BC Provincial-style卑诗省考风格 §7 Compound Interest复利 · BC PC12 / HSF-LE.A.3 [11 marks][11 分]

Two investment options are available for an initial principal of CA$$5{,}000$ held for $t$ years.本金 CA$$5{,}000$ 可选择两种投资方案，持有 $t$ 年。

Option A pays $6\%$ annual interest compounded monthly: $A(t) = 5000 \left(1 + \tfrac{0.06}{12}\right)^{12 t}$.
Option B is a linear-growth bond paying simple interest of CA$$320$ per year: $B(t) = 5000 + 320 \, t$.方案 A：年利率 $6\%$，按月复利：$A(t) = 5000 \left(1 + \tfrac{0.06}{12}\right)^{12 t}$。
方案 B：线性增长债券，按单利每年支付 CA$$320$：$B(t) = 5000 + 320 \, t$。

(a) Compute $A(5)$ and $B(5)$, each to the nearest cent. Which is larger after $5$ years, and by how much?计算 $A(5)$ 与 $B(5)$，各精确到分。$5$ 年后哪个更多？多多少？ [3]
(b) Determine the doubling time for Option A: the value of $t$ for which $A(t) = 10{,}000$. Use logarithms and give an exact expression, then round to two decimal places.确定方案 A 的倍增时间：使 $A(t) = 10{,}000$ 的 $t$ 值。用对数给出精确表达式，再保留两位小数。 [3]
(c) Cite HSF-LE.A.3: explain in one or two sentences why $A(t)$ must eventually exceed $B(t)$ regardless of how large the initial gap might be, and identify (approximately, by table or by inspection) the year $t$ in which $A(t)$ first exceeds $B(t)$.援引 HSF-LE.A.3，用一两句话说明为何无论初始差距多大，$A(t)$ 最终都会超过 $B(t)$，并通过列表或观察大致确定 $A(t)$ 首次超过 $B(t)$ 的年份 $t$。 [3]
(d) State one situational restriction on $t$ (e.g. non-negativity or a contractual maturity), and one reason a real saver might still prefer Option B in the short term.写出 $t$ 的一个情境限制（如非负性或合同到期日），并给出实际储户在短期内仍可能偏好方案 B 的一个理由。 [2]
🇺🇸 US Common Core美国共同核心HSF-LE.A.1 · HSF-LE.A.1c · HSF-LE.A.2 · HSF-LE.A.3 · HSF-LE.A.4 · HSF-LE.B.5 · HSF-IF.C.7e · HSF-BF.B.5 (+)
🇨🇦 Ontario安大略MCR3U Strand B Exponential Functions (expectation B2.1) · MCR3U Strand C geometric sequences / compound interest · MHF4U Exponential and Logarithmic Functions strand (laws of logs, log equations, inverse-of-exponential, base $e$)MCR3U 单元 B 指数函数（期望 B2.1）· MCR3U 单元 C 等比数列 / 复利 · MHF4U 指数函数与对数函数单元（对数运算律、对数方程、指数反函数、底数 $e$）
🇨🇦 British Columbia不列颠哥伦比亚PC12 Big Idea "Using inverses is the foundation of solving equations" · Content: exponential functions and equations (incl. base $e$); logarithms: operations, functions, and equations (common & natural logs, inverse of exponential); geometric sequences connecting to exponential functionsPC12 大概念"使用反函数是解方程的基础" · 内容：指数函数与方程（含底数 $e$）；对数：运算、函数与方程（常用对数与自然对数、指数的反函数）；等比数列与指数函数的衔接

Full 3-column Syllabus Map lives in ../Study Guides/Unit_5_Exponential_and_Logarithmic_Functions.html.完整的三列大纲对照表见 ../Study Guides/Unit_5_Exponential_and_Logarithmic_Functions.html。