AP Precalculus Unit 2: Exponential and Logarithmic Functions

Arithmetic and geometric sequences as the discrete versions of linear and exponential functions; the form $f(x)=ab^x$ and how to rewrite it; building exponential models and judging them with residuals; composition and inverse functions; logarithms as exponents and as the inverses of exponential functions; log graphs and log laws; solving exponential and logarithmic equations and inequalities; logarithmic models; and semi-log plots for spotting and fitting exponential data.

CED topics in this unit: 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15.

Key formulas and facts

• Arithmetic sequence: $a_n=a_0+dn$ (equal differences). Geometric sequence: $g_n=g_0r^n$ (equal ratios). Over equal input steps, a linear function adds a constant and an exponential function multiplies by a constant.
• $f(x)=ab^x$ with $a>0$: growth if $b>1$, decay if $0<b<1$; domain all reals; range $(0,\infty)$; horizontal asymptote $y=0$; always concave up.
• Rewriting: $b^{x+k}=b^k\cdot b^x$ (a horizontal shift is a vertical dilation); $b^{x/k}=\left(b^{1/k}\right)^x$ (changes the time unit of a growth factor).
• Percent change $r$ per unit: factor $1+r$. Half-life $h$: $a\left(\frac12\right)^{t/h}$. Continuous growth: $ae^{kt}$.
• Residual $=$ observed value $-$ predicted value. A good model has small residuals with no pattern.
• Composition: $(f\circ g)(x)=f(g(x))$; usually $f\circ g\ne g\circ f$. Inverse: $f\left(f^{-1}(x)\right)=x$; the graph of $f^{-1}$ is the graph of $f$ reflected in $y=x$.
• $\log_bc=a$ means $b^a=c$. With no base, $\log$ means base $10$; $\ln$ means base $e$.
• Log laws: $\log_b(mn)=\log_bm+\log_bn$, $\log_b\frac mn=\log_bm-\log_bn$, $\log_b\left(m^k\right)=k\log_bm$, $\log_bc=\frac{\ln c}{\ln b}$.
• $f(x)=\log_bx$ with $b>1$: domain $(0,\infty)$, range all reals, vertical asymptote $x=0$, increasing and concave down.
• Semi-log plot: if $y=ab^x$ then $\ln y=\ln a+x\ln b$, a line with slope $\ln b$ and intercept $\ln a$.

Worked examples

• Geometric terms: if $g_2=20$ and $g_5=160$, then $r^3=8$, $r=2$ and $g_0=5$, so $g_n=5\cdot2^n$.
• Exponential model from two points: through $(1,6)$ and $(3,54)$, $b^2=9$, $b=3$, $a=2$: $y=2\cdot3^x$.
• Changing the time unit: $500(1.08)^t$ with $t$ in years has monthly factor $1.08^{1/12}\approx1.0064$.
• Inverse: $f(x)=\frac{x-4}{2}$ has $f^{-1}(x)=2x+4$; check $f(f^{-1}(x))=\frac{2x+4-4}{2}=x$.
• Log equation with a check: $\log_2x+\log_2(x-2)=3$ gives $x^2-2x-8=0$, so $x=4$ or $x=-2$; only $x=4$ is in the domain.
• Exponential equation: $3e^{2t}=15$ gives $t=\frac{\ln5}{2}\approx0.805$.
• Inequality: $0.5^x<\frac18$ means $x>3$, because $0.5^x$ is decreasing.
• Semi-log line to model: $\ln y=0.7x+2$ gives $y=e^2\left(e^{0.7}\right)^x\approx7.389(2.014)^x$.

Common traps

• Writing $\log(a+b)$ as $\log a+\log b$: the product law is $\log(ab)=\log a+\log b$.
• Not rejecting a solution that makes a log's input zero or negative.
• Using $1+\frac{r}{12}$ for a monthly factor when the annual factor is $1+r$: the monthly factor is $(1+r)^{1/12}$.
• Reading $f^{-1}(x)$ as $\frac{1}{f(x)}$.
• Composing in the wrong order: $f(g(x))$ means apply $g$ first.
• Mixing up the sign of a residual: positive means the model is below the data (it underestimates).
• Forgetting that $\log$ on its own means base $10$ on the AP exam, while $\ln$ is base $e$.

Practice questions

148 multiple-choice and 4 free-response questions, each tagged with its CED topics and with "No calculator" or "Graphing calculator". Try each one before opening the answer. Each line of a free-response scoring guide is one point, and every free-response question here is out of 6. 59 of the 148 multiple-choice questions here need a graphing calculator; on the May 2027 exam, Part A of the multiple choice is 29 questions with no calculator and Part B is 13 questions with a graphing calculator. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

An arithmetic sequence has first term $a_1=7$ and common difference $4$. What is $a_{20}$?

1. $87$
2. $80$
3. $79$
4. $83$

Question 2 · Multiple choice

A geometric sequence has $g_1=6$ and $g_4=162$. What is $g_6$?

1. $1458$
2. $486$
3. $4374$
4. $972$

Question 3 · Multiple choice

In an arithmetic sequence, $a_3=11$ and $a_{10}=46$. Which expression gives $a_n$?

1. $5n+1$
2. $7n-10$
3. $5n-4$
4. $5n-9$

Question 4 · Multiple choice

The sequence $400, 300, 225, \ldots$ is geometric, with $a_1=400$. Which expression gives $a_n$?

1. $400\left(0.75\right)^{n}$
2. $400\left(0.75\right)^{n-1}$
3. $500-100n$
4. $400\left(1.25\right)^{n-1}$

Question 5 · Multiple choice

Which list could be the first four terms of a geometric sequence?

1. $5,\ 10,\ 15,\ 20$
2. $5,\ -10,\ 20,\ -40$
3. $5,\ 10,\ 20,\ 30$
4. $5,\ -5,\ -15,\ -25$

Question 6 · Multiple choice

An arithmetic sequence has $a_0=12$ and $a_5=-3$. What is $a_{12}$?

1. $-36$
2. $-21$
3. $-27$
4. $-24$

Question 7 · Multiple choice

A geometric sequence has $g_2=12$ and $g_5=-96$. What is $g_0$?

1. $3$
2. $-3$
3. $6$
4. $-6$

Question 8 · Multiple choice

Account A starts with $\$1000$ and gains $\$50$ each month. Account B starts with $\$1000$ and grows by $2\%$ each month. After $24$ months, how much more is in account A than in account B, in dollars?

1. $608.437$
2. $1200.000$
3. $591.563$
4. $408.437$

Question 9 · Multiple choice

A geometric sequence has $g_1=3.2$ and common ratio $1.15$. To three decimal places, what is $g_{12}$?

1. $17.121$
2. $14.888$
3. $4.850$
4. $40.480$

Question 10 · Multiple choice

Sequence $a$ is arithmetic with $a_1=120$ and common difference $9$. Sequence $g$ is geometric with $g_1=50$ and common ratio $1.06$. What is the least $n$ for which $g_n>a_n$?

1. $39$
2. $41$
3. $40$
4. $43$

Question 11 · Multiple choice

A geometric sequence has $g_1=800$ and $g_n=800(0.93)^{n-1}$. What is the first term number $n$ for which $g_n<100$?

1. $29$
2. $31$
3. $28$
4. $30$

Question 12 · Multiple choice

Which type of function fits the data in the table exactly?

1. Linear
2. Exponential
3. Quadratic
4. Cubic

Question 13 · Multiple choice

The table shows values of a function that changes by equal factors over equal-length input intervals. Which rule fits the table?

1. $80\left(0.9\right)^{x}$
2. $80-4x$
3. $80\left(0.9\right)^{x/2}$
4. $80\left(0.95\right)^{x}$

Question 14 · Multiple choice

A linear function $f$ and an exponential function $g$ both have value $4$ at $x=0$ and value $12$ at $x=1$. What is $g(3)-f(3)$?

1. $80$
2. $0$
3. $64$
4. $108$

Question 15 · Multiple choice

An exponential function $h$ has $h(1)=10$ and $h(3)=40$. What is $h(5)$?

1. $70$
2. $80$
3. $640$
4. $160$

Question 16 · Multiple choice

What is the least positive integer $x$ for which $1.5^x>50x$?

1. $16$
2. $17$
3. $15$
4. $18$

Question 17 · Multiple choice

The sequence $3, 7, 11, 15, \ldots$ is written as a function $f(n)$ with $f(0)=3$. Which is $f(n)$?

1. $4n+3$
2. $3n+4$
3. $3\cdot 4^{n}$
4. $4n-1$

Question 18 · Multiple choice

A quantity grows by $20\%$ each year. By what percent does it grow over any $2$-year period?

1. $40\%$
2. $20\%$
3. $24\%$
4. $44\%$

Question 19 · Multiple choice

Let $f(x)=500+40x$ and $g(x)=200(1.08)^x$. For $x>0$, the graphs intersect once. To three decimal places, what is the $x$-coordinate of the intersection?

1. $26.785$
2. $7.500$
3. $11.906$
4. $-10.224$

Question 20 · Multiple choice

What is the least positive integer $n$ for which $1.03^n>1+0.05n$?

1. $32$
2. $33$
3. $34$
4. $28$

Question 21 · Multiple choice

Two savings plans both start at $\$1200$. Plan A adds $\$40$ each month; Plan B grows by $2.5\%$ each month. The balances are modeled by $A(t)=1200+40t$ and $B(t)=1200(1.025)^t$. To three decimal places, at what positive time $t$, in months, are the balances equal?

1. $12.152$
2. $28.071$
3. $23.199$
4. $24.000$

Question 22 · Multiple choice

Which statement describes $f(x)=3(0.5)^x$?

1. $f$ is decreasing and its graph is concave down
2. $f$ is increasing and its graph is concave up
3. $f$ is decreasing and its graph is concave up
4. $f$ is increasing and its graph is concave down

Question 23 · Multiple choice

What is $\displaystyle\lim_{x\to-\infty}\left(4\cdot2^x+1\right)$?

1. $0$
2. $1$
3. $4$
4. $-\infty$

Question 24 · Multiple choice

The graph of an exponential function $f(x)=ab^x$ is shown. Which of the following is $f(x)$?

1. $3\cdot 2^{x}$
2. $2\cdot 6^{x}$
3. $2\cdot 3^{x}$
4. $6\cdot 3^{x}$

Question 25 · Multiple choice

What is the range of $g(x)=-2\cdot5^x+3$?

1. $(3,\infty)$
2. $(-\infty,1)$
3. $(-\infty,\infty)$
4. $(-\infty,3)$

Question 26 · Multiple choice

Let $f(x)=7(1.2)^{-x}$. When $x$ increases by $1$, the value of $f(x)$ is multiplied by

1. $\frac{5}{6}$
2. $1.2$
3. $-1.2$
4. $0.2$

Question 27 · Multiple choice

An exponential function $f(x)=ab^x$ with $b>0$ has $f(0)=5$ and $f(2)=45$. What is $f(-1)$?

1. $\frac{5}{3}$
2. $15$
3. $\frac{1}{3}$
4. $-15$

Question 28 · Multiple choice

Let $f(x)=3(0.8)^x+2$. To three decimal places, what is $f(4.6)$?

1. $13.040$
2. $1.075$
3. $1.791$
4. $3.075$

Question 29 · Multiple choice

Let $f(x)=150(1.12)^x$. To three decimal places, what is the average rate of change of $f$ over $[2.5,6]$?

1. $27.698$
2. $96.944$
3. $16.157$
4. $18.000$

Question 30 · Multiple choice

A cup of tea cools so that its temperature above room temperature, in degrees, is modeled by $g(t)=12e^{-0.35t}+4$, where $t$ is in minutes. To three decimal places, what is $g(2.2)$?

1. $5.556$
2. $9.556$
3. $7.408$
4. $29.917$

Question 31 · Multiple choice

Which expression is equivalent to $8^{2x/3}$?

1. $16^{x}$
2. $2^{x}$
3. $4^{x}$
4. $\left(\frac{16}{3}\right)^{x}$

Question 32 · Multiple choice

Which expression is equivalent to $5\cdot2^{x+3}$?

1. $10\cdot 2^{x}$
2. $5\cdot 2^{x}+8$
3. $10^{x+3}$
4. $40\cdot 2^{x}$

Question 33 · Multiple choice

Which expression is equivalent to $\left(3^{x}\right)^2\cdot3^{-x+1}$?

1. $3^{2x}$
2. $3^{x+1}$
3. $3^{-2x^2+2x}$
4. $9^{x+1}$

Question 34 · Multiple choice

Which expression is equivalent to $\dfrac{e^{2x}}{e^{x-3}}$?

1. $e^{x-3}$
2. $e^{x+3}$
3. $e^{3x-3}$
4. $e^{2}$

Question 35 · Multiple choice

Let $g(x)=2^x$. For every $x$, $g(x+4)=k\cdot g(x)$. What is $k$?

1. $16$
2. $4$
3. $8$
4. $6$

Question 36 · Multiple choice

Which expression is equivalent to $100(0.81)^{t/2}$?

1. $100\left(0.405\right)^{t}$
2. $50\left(0.81\right)^{t}$
3. $100\left(0.81\right)^{2t}$
4. $100\left(0.9\right)^{t}$

Question 37 · Multiple choice

The expression $7(1.5)^{x/3}$ can be written as $7b^x$. To three decimal places, what is $b$?

1. $0.500$
2. $1.500$
3. $1.145$
4. $3.375$

Question 38 · Multiple choice

The model $A(t)=200e^{0.06t}$ is rewritten in the form $A(t)=200b^t$. To three decimal places, what is $b$?

1. $1.060$
2. $0.942$
3. $1.124$
4. $1.062$

Question 39 · Multiple choice

A substance has a half-life of $5.3$ years. Its amount is modeled by $A(t)=A_0b^t$, with $t$ in years. To three decimal places, what is $b$?

1. $0.877$
2. $0.906$
3. $0.189$
4. $1.140$

Question 40 · Multiple choice

A town has $2400$ residents, and its population grows by $3.5\%$ each year. Which function gives the population $t$ years from now?

1. $2400\left(0.035\right)^{t}$
2. $2400\left(1.35\right)^{t}$
3. $2400\left(1.035\right)^{t}$
4. $2400+3.5t$

Question 41 · Multiple choice

A medicine has a half-life of $6$ hours. A patient has $80$ mg in the body at $t=0$. Which function gives the amount, in mg, after $t$ hours?

1. $80\left(\frac{1}{2}\right)^{6t}$
2. $80\left(\frac{1}{2}\right)^{t/6}$
3. $80-6t$
4. $40\left(\frac{1}{2}\right)^{t}$

Question 42 · Multiple choice

A car loses $12\%$ of its value each year. To the nearest thousandth of a year, how long does it take to lose half its value?

1. $4.167$
2. $6.000$
3. $5.776$
4. $5.422$

Question 43 · Multiple choice

A quantity has an annual growth factor of $1.2$. To the nearest thousandth, what is its monthly growth factor?

1. $1.017$
2. $1.100$
3. $1.015$
4. $1.200$

Question 44 · Multiple choice

An account balance in dollars is modeled by $A(t)=500e^{0.04t}$, where $t$ is years. What is $A(10)$?

1. $745.912$
2. $700.000$
3. $740.122$
4. $520.404$

Question 45 · Multiple choice

An exponential model $y=ab^x$ passes through $(2,18)$ and $(5,486)$. Which is the model?

1. $3\cdot 2^{x}$
2. $18\cdot 3^{x}$
3. $2\cdot 27^{x}$
4. $2\cdot 3^{x}$

Question 46 · Multiple choice

Use an exponential regression $y=ab^x$ on the data in the table to predict $y$ when $x=7$.

1. $33.891$
2. $59.815$
3. $60.080$
4. $40.723$

Question 47 · Multiple choice

An exponential model $y=ab^x$ passes through $(1.5,\,20)$ and $(4,\,52)$. To three decimal places, what is $b$?

1. $1.040$
2. $2.600$
3. $1.466$
4. $1.270$

Question 48 · Multiple choice

The table shows the number of members of an online club $x$ weeks after it opened. Use an exponential regression $y=ab^x$ to find the model's doubling time, in weeks.

1. $1.777$
2. $5.516$
3. $8.000$
4. $5.855$

Question 49 · Multiple choice

A model $y=2x+3$ is used for a data set. At $x=4$ the observed value is $12.5$. What is the residual at $x=4$?

1. $1.5$
2. $-1.5$
3. $11$
4. $12.5$

Question 50 · Multiple choice

A linear regression line is fit to the data in the table. What are the signs of its residuals at $x=0$, $x=2$ and $x=4$, in that order?

1. $-,\ +,\ -$
2. $+,\ -,\ +$
3. $+,\ +,\ +$
4. $-,\ -,\ -$

Question 51 · Multiple choice

Two models are proposed for the data in the table: Model A, $y=8x-5$, and Model B, $y=2.4(1.9)^x$. Based on the sum of the squared residuals, which model fits the data better?

1. Model A: its sum of squared residuals is smaller
2. Model A: it is linear
3. Model B: its sum of squared residuals is smaller
4. Both fit equally well

Question 52 · Multiple choice

Use an exponential regression $y=ab^x$ on the data in the table to predict $y$ at $x=6$.

1. $20.300$
2. $15.000$
3. $24.680$
4. $22.265$

Question 53 · Multiple choice

At one data point the observed value is $40$ and the residual is $-3$. What value did the model predict?

1. $43$
2. $37$
3. $-43$
4. $3$

Question 54 · Multiple choice

The model $y=3x+4$ is used for a data set. At $x=5$ the observed value is $20$. Which statement is true?

1. The model overestimates by $1$
2. The model underestimates by $1$
3. The model underestimates by $19$
4. The model overestimates by $20$

Question 55 · Multiple choice

A student fits both a linear and an exponential model to the data in the table. Which model is more appropriate?

1. Linear, because the outputs increase
2. Linear, because the first differences are constant
3. Exponential, because the ratios of consecutive outputs are constant
4. Neither, because the outputs double

Question 56 · Multiple choice

An exponential regression $y=ab^x$ is fit to the data in the table. To three decimal places, what is the residual at $x=3$?

1. $0.220$
2. $-0.220$
3. $8.380$
4. $-1.880$

Question 57 · Multiple choice

A linear regression is fit to the data in the table. To three decimal places, what is the residual at $x=4$?

1. $0.343$
2. $-0.343$
3. $9.143$
4. $-0.180$

Question 58 · Multiple choice

A quadratic regression is fit to the data in the table. To three decimal places, what is the residual at $x=2$?

1. $-0.080$
2. $3.420$
3. $0.080$
4. $-1.396$

Question 59 · Multiple choice

With $f(x)=2x^2-3$ and $g(x)=x+4$, find $f(g(2))$.

1. $69$
2. $9$
3. $21$
4. $75$

Question 60 · Multiple choice

The table gives values of $f$ and $g$. What is $g(f(1))$?

1. $2$
2. $3$
3. $4$
4. $1$

Question 61 · Multiple choice

Let $f(x)=\sqrt{x}$ and $g(x)=x^2+9$. Which is $(f\circ g)(x)$?

1. $x+3$
2. $\sqrt{x^2+9}$
3. $x+9$
4. $\sqrt{x}+9$

Question 62 · Multiple choice

Let $h(x)=(2x-5)^3$ and $g(x)=2x-5$. Which function $f$ gives $h(x)=f(g(x))$?

1. $(2x-5)^{3}$
2. $2x^{3}-5$
3. $(x-5)^{3}$
4. $x^{3}$

Question 63 · Multiple choice

For $f(x)=3^x$ and $g(x)=x+2$, which expression equals $f(g(x))$?

1. $3^{x}+2$
2. $2\cdot3^{x}$
3. $9\cdot3^{x}$
4. $3^{2x}$

Question 64 · Multiple choice

Let $f(x)=\sqrt{x}$ and $g(x)=6-2x$. What is the domain of $f(g(x))$?

1. $[3,\infty)$
2. $(-\infty,3]$
3. $[0,\infty)$
4. $[0,3]$

Question 65 · Multiple choice

The graph of $f$ is shown, and the table gives values of $g$. What is $g(f(2))$?

1. $7$
2. $-1$
3. $5$
4. $0$

Question 66 · Multiple choice

Let $g(x)=x^2-2.5$ and $f(x)=e^{0.4x}$. To three decimal places, what is $f\big(g(1.7)\big)$?

1. $1.396$
2. $0.770$
3. $2.364$
4. $1.169$

Question 67 · Multiple choice

Let $g(x)=x^2+3x$ and $f(x)=\ln(x+1)$. The equation $g\big(f(x)\big)=10$ has one positive solution. To three decimal places, what is it?

1. $6.389$
2. $-0.993$
3. $2.000$
4. $22.624$

Question 68 · Multiple choice

Let $f(x)=2^x$ and $g(x)=\sqrt{x+1}$. To three decimal places, what is $f\big(g(2.4)\big)$?

1. $2.506$
2. $3.590$
3. $9.732$
4. $10.556$

Question 69 · Multiple choice

Let $f(x)=\dfrac{x+5}{3}$. Which is $f^{-1}(x)$?

1. $3x+5$
2. $\frac{x-5}{3}$
3. $\frac{3}{x+5}$
4. $3x-5$

Question 70 · Multiple choice

The table gives values of an invertible function $f$. What is $f^{-1}(13)$?

1. $3$
2. $13$
3. $\frac{1}{13}$
4. $21$

Question 71 · Multiple choice

Let $f(x)=x^3-2$. What is $f^{-1}(25)$?

1. $\sqrt[3]{23}$
2. $15623$
3. $3$
4. $\frac{1}{25}$

Question 72 · Multiple choice

On which of these domains is $f(x)=(x-2)^2$ one-to-one?

1. $[0,\infty)$
2. $[2,\infty)$
3. $(-\infty,\infty)$
4. $[-2,\infty)$

Question 73 · Multiple choice

The graph of an invertible function $f$ is shown. Which point is on the graph of $f^{-1}$?

1. $\left(1,4\right)$
2. $\left(-1,-4\right)$
3. $\left(\frac{1}{4},1\right)$
4. $\left(4,1\right)$

Question 74 · Multiple choice

Let $f(x)=4x+1$. Which function $g$ satisfies $f(g(x))=x$ for all $x$?

1. $\frac{x+1}{4}$
2. $4x-1$
3. $\frac{1}{4x+1}$
4. $\frac{x-1}{4}$

Question 75 · Multiple choice

Let $f(x)=\dfrac{2x+1}{x-3}$. Which is $f^{-1}(x)$?

1. $\frac{x-3}{2x+1}$
2. $\frac{3x+1}{x-2}$
3. $\frac{3x+1}{x+2}$
4. $\frac{2x-1}{x+3}$

Question 76 · Multiple choice

Let $f(x)=x^3+x+1$, which is increasing and so has an inverse. To three decimal places, what is $f^{-1}(5)$?

1. $0.008$
2. $1.587$
3. $1.379$
4. $1.710$

Question 77 · Multiple choice

Let $f(x)=2e^x+x$. To three decimal places, what is $f^{-1}(6)$?

1. $1.099$
2. $0.001$
3. $0.099$
4. $0.930$

Question 78 · Multiple choice

Let $g(x)=x^5+2x^3+3$. To three decimal places, what is $g^{-1}(10)$?

1. $1.252$
2. $1.476$
3. $1.913$
4. $1.670$

Question 79 · Multiple choice

What is $\log_2 32$?

1. $16$
2. $\frac{1}{5}$
3. $5$
4. $6$

Question 80 · Multiple choice

What is $\log_3\left(\frac{1}{81}\right)$?

1. $-4$
2. $4$
3. $-\frac{1}{4}$
4. $-27$

Question 81 · Multiple choice

What is $\log 0.001$?

1. $3$
2. $-3$
3. $-\frac{1}{3}$
4. $-2$

Question 82 · Multiple choice

$\log_5 40$ lies strictly between which pair of consecutive whole numbers?

1. $2$ and $3$
2. $1$ and $2$
3. $3$ and $4$
4. $8$ and $9$

Question 83 · Multiple choice

What is $\ln\sqrt{e}$?

1. $2$
2. $\sqrt{e}$
3. $\frac{1}{2}$
4. $1$

Question 84 · Multiple choice

What is $\log_4 8$?

1. $2$
2. $\frac{2}{3}$
3. $\frac{1}{2}$
4. $\frac{3}{2}$

Question 85 · Multiple choice

If $\log_b 64=3$, what is $b$?

1. $8$
2. $\frac{64}{3}$
3. $4$
4. $192$

Question 86 · Multiple choice

What is $\log_7 50$, to three decimal places?

1. $1.699$
2. $0.497$
3. $7.143$
4. $2.010$

Question 87 · Multiple choice

To three decimal places, what is $\log_{1.5}12$?

1. $0.903$
2. $6.129$
3. $8.000$
4. $0.163$

Question 88 · Multiple choice

To three decimal places, what is $\log_4 0.3$?

1. $0.868$
2. $-0.131$
3. $-0.868$
4. $0.740$

Question 89 · Multiple choice

Let $f(x)=\log_3 x$. Which is $f^{-1}(x)$?

1. $x^{3}$
2. $3^{x}$
3. $\left(\frac{1}{3}\right)^{x}$
4. $\log_{x}(3)$

Question 90 · Multiple choice

The point $(2,25)$ is on the graph of $y=5^x$. Which point must be on the graph of $y=\log_5x$?

1. $\left(25,2\right)$
2. $\left(2,25\right)$
3. $\left(-2,25\right)$
4. $\left(25,-2\right)$

Question 91 · Multiple choice

Solve $10^{\log(x+2)}=7$.

1. $x=7$
2. $x=10^{7}-2$
3. $x=5$
4. $x=\log 7-2$

Question 92 · Multiple choice

What is $e^{\ln4+\ln3}$?

1. $7$
2. $e^{7}$
3. $\ln 12$
4. $12$

Question 93 · Multiple choice

The graph of $y=2^x$ is shown with its reflection $g$ over the line $y=x$ (dashed). What is $g(16)$?

1. $4$
2. $65536$
3. $\frac{1}{16}$
4. $8$

Question 94 · Multiple choice

Let $f(x)=3^x$. What is $f^{-1}\left(\frac{1}{9}\right)$?

1. $2$
2. $-2$
3. $\frac{1}{2}$
4. $-\frac{1}{2}$

Question 95 · Multiple choice

Let $f(x)=4\cdot3^x$. To three decimal places, what is $f^{-1}(50)$?

1. $3.561$
2. $4.167$
3. $1.574$
4. $2.299$

Question 96 · Multiple choice

Let $f(x)=\ln(x+2)-1$. To three decimal places, what is $f^{-1}(0.8)$?

1. $4.050$
2. $1.226$
3. $0.030$
4. $8.050$

Question 97 · Multiple choice

Let $g(x)=\log_2(3x-1)$. To three decimal places, what is $g^{-1}(4.5)$?

1. $7.209$
2. $7.876$
3. $3.644$
4. $7.542$

Question 98 · Multiple choice

What is the domain of $g(x)=\ln(5-x)$?

1. $(5,\infty)$
2. $(-\infty,5]$
3. $(0,\infty)$
4. $(-\infty,5)$

Question 99 · Multiple choice

The curve $y=\log_2(x+3)-1$ has one vertical asymptote. Which line is it?

1. $x=3$
2. $x=-1$
3. $x=-3$
4. $x=1$

Question 100 · Multiple choice

The graph of a logarithmic function $f$ is shown, with its vertical asymptote dashed. Which of the following could be $f(x)$?

1. $\log_{2}(x+1)$
2. $\log_{2}(x-1)$
3. $\log_{2}(x)-1$
4. $-\log_{2}(x-1)$

Question 101 · Multiple choice

On its whole domain, how does the graph of $y=\ln x$ behave?

1. It rises, and it bends downward (concave down)
2. It rises, and it bends upward (concave up)
3. It falls, and it bends downward (concave down)
4. It falls, and it bends upward (concave up)

Question 102 · Multiple choice

What is $\displaystyle\lim_{x\to0^+}\log_3x$?

1. $0$
2. $\infty$
3. $-\infty$
4. $1$

Question 103 · Multiple choice

What is the range of $y=4\log x$ on the domain $(0,1000]$?

1. $(0,12]$
2. $(-\infty,12]$
3. $(-\infty,4000]$
4. $[0,12]$

Question 104 · Multiple choice

Where does the graph of $y=\ln(2x-3)$ cross the $x$-axis?

1. $x=\frac{3}{2}$
2. $x=1$
3. $x=\frac{1}{2}$
4. $x=2$

Question 105 · Multiple choice

For which value of $b$ is $f(x)=\log_b x$ a decreasing function?

1. $b=0.3$
2. $b=3$
3. $b=e$
4. $b=10$

Question 106 · Multiple choice

Let $f(x)=2\ln x-\dfrac{x}{3}$ for $x>0$. To three decimal places, what is the maximum value of $f$?

1. $6.000$
2. $3.584$
3. $1.584$
4. $-0.208$

Question 107 · Multiple choice

To three decimal places, what is the zero of $h(x)=\ln x+x-3$?

1. $20.086$
2. $2.000$
3. $2.718$
4. $2.208$

Question 108 · Multiple choice

The function $f(x)=\log x-0.1x+1$ has two zeros. To three decimal places, what is the greater one?

1. $23.758$
2. $0.102$
3. $4.343$
4. $10.000$

Question 109 · Multiple choice

What is $\log_2 24-\log_2 3$?

1. $3$
2. $\log_{2}(21)$
3. $8$
4. $\frac{\log_{2}(24)}{\log_{2}(3)}$

Question 110 · Multiple choice

Which expression is equivalent to $2\ln x+\ln3-\ln y$ for $x>0$, $y>0$?

1. $\ln\left(\frac{6x}{y}\right)$
2. $\ln\left(3x^2-y\right)$
3. $\ln\left(\frac{x^2+3}{y}\right)$
4. $\ln\left(\frac{3x^2}{y}\right)$

Question 111 · Multiple choice

If $\log_3 a=2.1$ and $\log_3 b=0.4$, what is $\log_3\left(a^2b\right)$?

1. $0.84$
2. $2.5$
3. $4.6$
4. $4.2$

Question 112 · Multiple choice

What is $\log_8 32$?

1. $4$
2. $\frac{5}{3}$
3. $\frac{3}{5}$
4. $\frac{1}{4}$

Question 113 · Multiple choice

The graph of $y=\log_2(8x)$ is the graph of $y=\log_2x$ shifted vertically by how many units?

1. $3$
2. $8$
3. $\frac{1}{8}$
4. $-3$

Question 114 · Multiple choice

Which expression is equivalent to $\ln\left(e^2x^3\right)$ for $x>0$?

1. $6\ln x$
2. $2\ln x^{3}+1$
3. $3+2\ln x$
4. $2+3\ln x$

Question 115 · Multiple choice

What is $\log(1000x)-\log\left(\frac{x}{100}\right)$ for $x>0$?

1. $1$
2. $5$
3. $3$
4. $100000$

Question 116 · Multiple choice

For all $x>0$, $\log_2x=k\ln x$ for a constant $k$. To three decimal places, what is $k$?

1. $0.693$
2. $1.443$
3. $0.301$
4. $3.322$

Question 117 · Multiple choice

For all $x>0$, $\log_5x=k\log x$ (where $\log$ is base $10$). To three decimal places, what is $k$?

1. $0.699$
2. $1.609$
3. $1.431$
4. $0.621$

Question 118 · Multiple choice

Solve $2^{x+1}=32$.

1. $x=5$
2. $x=15$
3. $x=4$
4. $x=16$

Question 119 · Multiple choice

Solve $5e^{0.3t}=40$ for $t$. Give your answer to three decimal places.

1. $2.079$
2. $7.000$
3. $6.931$
4. $26.667$

Question 120 · Multiple choice

What is the solution set of $\log_3(x-1)+\log_3(x+1)=1$?

1. $\{2\}$
2. $\{-2,2\}$
3. $\{-2\}$
4. No solution

Question 121 · Multiple choice

For which $x$ is $3^x>81$?

1. $(-\infty,4)$
2. $(4,\infty)$
3. $(27,\infty)$
4. $(3,\infty)$

Question 122 · Multiple choice

For which $x$ is $\ln x\le2$?

1. $(-\infty,e^{2}]$
2. $(0,2]$
3. $[e^{2},\infty)$
4. $(0,e^{2}]$

Question 123 · Multiple choice

Let $f(x)=4e^x-1$. Which is $f^{-1}(x)$?

1. $\ln\left(4x-1\right)$
2. $\frac{\ln(x+1)}{4}$
3. $4\ln x+1$
4. $\ln\left(\frac{x+1}{4}\right)$

Question 124 · Multiple choice

To three decimal places, what is the solution of $3^{x+1}=7^{x-2}$?

1. $3.297$
2. $3.593$
3. $3.000$
4. $5.890$

Question 125 · Multiple choice

The equation $2^x=x+4$ has one positive solution. To three decimal places, what is it?

1. $2.756$
2. $2.000$
3. $-3.935$
4. $3.000$

Question 126 · Multiple choice

To three decimal places, what is the solution of $\ln x+\ln(x+3)=2$?

1. $4.389$
2. $1.605$
3. $-4.605$
4. $3.695$

Question 127 · Multiple choice

The solution of the inequality $5e^{0.2x}>40-x$ is $x>c$. To three decimal places, what is $c$?

1. $10.397$
2. $26.409$
3. $9.106$
4. $11.678$

Question 128 · Multiple choice

The loudness of a sound in decibels is $L=10\log\left(\frac{I}{I_0}\right)$, where $I$ is its intensity. If the intensity is multiplied by $100$, how does $L$ change?

1. It increases by $20$ decibels
2. It is multiplied by $100$
3. It increases by $100$ decibels
4. It increases by $2$ decibels

Question 129 · Multiple choice

The pH of a solution is $-\log\left[\mathrm{H}^+\right]$. What is the pH when $\left[\mathrm{H}^+\right]=3.2\times10^{-5}$?

1. $5.000$
2. $3.200$
3. $4.495$
4. $4.505$

Question 130 · Multiple choice

The height of a plant, in centimeters, $x$ weeks after planting is modeled by $h(x)=12+5\ln x$ for $x\ge1$. What is $h(8)$?

1. $52.000$
2. $22.397$
3. $16.515$
4. $32.794$

Question 131 · Multiple choice

A model $y=a+b\ln x$ passes through $(1,7)$ and $\left(e^2,13\right)$. What is $b$?

1. $6$
2. $7$
3. $3$
4. $\frac{13}{7}$

Question 132 · Multiple choice

For the model $y=4+6\ln x$, by how much does $y$ increase whenever $x$ doubles?

1. $12$
2. $6$
3. $\ln 12$
4. $6\ln 2$

Question 133 · Multiple choice

Earthquake magnitude is modeled by $M=\log A+c$, where $A$ is the measured wave amplitude and $c$ is a constant. One earthquake has magnitude $6.2$ and another $4.2$. How many times larger is the first one's amplitude?

1. $100$
2. $2$
3. $20$
4. $1.476$

Question 134 · Multiple choice

Use a logarithmic regression $y=a+b\ln x$ on the data in the table to predict $y$ at $x=32$.

1. $12.700$
2. $12.960$
3. $13.850$
4. $11.920$

Question 135 · Multiple choice

The time, in years, for a population to grow from $2000$ to $P$ is modeled by $t(P)=\log_{1.05}\left(\frac{P}{2000}\right)$. What is $t(4000)$?

1. $20.000$
2. $0.301$
3. $14.207$
4. $40.000$

Question 136 · Multiple choice

Sound level in decibels is modeled by $L=10\log\left(\dfrac{I}{I_0}\right)$, where $I$ is the sound's intensity. A $73$-decibel sound is how many times as intense as a $61$-decibel sound?

1. $1.200$
2. $12.000$
3. $1.197$
4. $15.849$

Question 137 · Multiple choice

Use a logarithmic regression $y=a+b\ln x$ on the data in the table to predict $y$ when $x=24$.

1. $14.250$
2. $17.695$
3. $9.062$
4. $74.329$

Question 138 · Multiple choice

A model $y=a+b\ln x$ passes through $(2,\,9.4)$ and $(10,\,15.2)$. To three decimal places, what does the model predict at $x=30$?

1. $3.604$
2. $19.159$
3. $29.700$
4. $21.657$

Question 139 · Multiple choice

Data that fit $y=3\cdot2^x$ exactly are plotted with $\ln y$ on the vertical axis against $x$. What is the slope of the resulting line?

1. $2$
2. $\ln 3$
3. $3$
4. $\ln 2$

Question 140 · Multiple choice

A semi-log plot of a data set ($\ln y$ against $x$) is well fit by $\ln y=0.5x+1.2$. Which exponential model matches?

1. $1.2\left(0.5\right)^{x}$
2. $e^{1.2}\left(e^{0.5}\right)^{x}$
3. $0.5x+1.2$
4. $e^{0.5}\left(e^{1.2}\right)^{x}$

Question 141 · Multiple choice

The figure shows a semi-log plot of a data set, with $\log y$ (base $10$) on the vertical axis. The data lie on the line shown. What is $y$ when $x=2$?

1. $100$
2. $2$
3. $10$
4. $1000$

Question 142 · Multiple choice

Which data set, at $x=0,1,2,3$, would lie on a straight line on a semi-log plot ($\ln y$ against $x$)?

1. $2,\ 6,\ 10,\ 14$
2. $2,\ 4,\ 8,\ 14$
3. $2,\ 6,\ 18,\ 54$
4. $1,\ 4,\ 9,\ 16$

Question 143 · Multiple choice

Data are modeled by $y=5(1.3)^x$. Which expression gives $\ln y$?

1. $\ln(5)+x\ln(1.3)$
2. $5+x\ln(1.3)$
3. $\ln(5)\cdot x\ln(1.3)$
4. $x\ln(6.5)$

Question 144 · Multiple choice

For the data in the table, find the linear regression of $\ln y$ on $x$. To three decimal places, what is its slope?

1. $9.230$
2. $0.694$
3. $1.384$
4. $2.001$

Question 145 · Multiple choice

A semi-log plot of a data set ($\ln y$ against $x$) is well fit by $\ln y=-0.2x+3$. By what factor does the model's $y$ change when $x$ increases by $1$?

1. $-0.2$
2. $0.8$
3. $e^{3}$
4. $e^{-0.2}$

Question 146 · Multiple choice

For the data in the table, a line is fit to the semi-log plot ($\ln y$ against $x$) by linear regression. In the matching exponential model $y=ab^x$, what is $b$, to three decimal places?

1. $0.398$
2. $2.467$
3. $1.489$
4. $1.463$

Question 147 · Multiple choice

For the data in the table, a line is fit to the semi-log plot ($\ln y$ against $x$) by linear regression. In the matching model $y=ab^x$, what is the initial value $a$, to three decimal places?

1. $3.982$
2. $40.000$
3. $0.741$
4. $53.610$

Question 148 · Multiple choice

A semi-log plot of a data set ($\ln y$ against $x$) is well fit by the line $\ln y=0.32x+0.85$. To three decimal places, what $y$-value does the model predict at $x=7$?

1. $21.977$
2. $3.090$
3. $10.243$
4. $11.733$

Question 149 · Free response (6 points) · Function concepts

The table gives values of a function $g$. The function $f$ is given by $f(x)=2^{x-1}$. A graphing calculator is allowed.

1. (A(i)) Find $(f\circ g)(0)$. [1 point]
2. (A(ii)) Find $(g\circ f)(3)$. [1 point]
3. (B) Find an expression for $f^{-1}(x)$, and use it to find $f^{-1}(32)$. [2 points]
4. (C) Find the average rates of change of $f$ over $[1,3]$ and over $[3,5]$. Use them to say whether the graph of $f$ is concave up or concave down on $[1,5]$. [2 points]

Question 150 · Free response (6 points) · Modeling a non-periodic context

A cup of hot water cools in a room. The difference $D$, in degrees Fahrenheit, between the water's temperature and the room's temperature is $150$ at time $t=0$ minutes and $90$ at $t=10$ minutes. The difference is modeled by $D(t)=ab^t$. A graphing calculator is allowed.

1. (A(i)) Find $a$ and $b$. Give $b$ to three decimal places. [1 point]
2. (A(ii)) Use the model to find $D(25)$. [1 point]
3. (B) Find the time when the difference is $30$ degrees Fahrenheit. [2 points]
4. (C) (i) The model can be written $D(t)=150e^{kt}$. Find $k$ to three decimal places. (ii) Find $\displaystyle\lim_{t\to\infty}D(t)$ and explain what it means for the water's temperature. [2 points]

Question 151 · Free response (6 points) · Modeling a non-periodic context

The table gives the number of bacteria $y$, in thousands, in a culture $x$ hours after it was started. A graphing calculator is allowed.

1. (A(i)) Find the ratios of consecutive values of $y$, to three decimal places, and explain why an exponential model is reasonable. [1 point]
2. (A(ii)) Find the linear regression of $\ln y$ on $x$. Give the slope and intercept to three decimal places. [1 point]
3. (B(i)) Use your answer to (A)(ii) to write an exponential model $y=ab^x$. Give $a$ and $b$ to three decimal places. [1 point]
4. (B(ii)) Use the model to predict the number of bacteria at $x=6$. [1 point]
5. (C) Find the residual of the model at $x=4$. Does the model overestimate or underestimate the data there? [2 points]

Question 152 · Free response (6 points) · Symbolic manipulation

Answer each part without a calculator. Show your work.

1. (A(i)) Solve $3^{2x-1}=27$ for $x$. [1 point]
2. (A(ii)) Solve $\log_2(x+3)+\log_2(x-3)=4$ for $x$. [1 point]
3. (B(i)) Rewrite $2\log x-\log4+\log5$ as a single logarithm, for $x>0$. [1 point]
4. (B(ii)) Rewrite $8^{x/3}\cdot4^{x}$ in the form $2^{kx}$, and give $k$. [1 point]
5. (C) Let $h(x)=5+\ln(x-2)$ for $x>2$. Find $h^{-1}(x)$, and find $h^{-1}(5)$. [2 points]