Modeling with Exponential & Logarithmic Functions (College Board AP® Precalculus): Exam Questions

The variables $x$ and $y$ are related by the exponential equation $y=5·4^x$. Which of the following equations expresses $\log y$ as a linear function of $x$?

A model $M$ is constructed to predict the population of a town, in thousands. The model gives $M\left(5\right)=12.4$ for year $t=5$. The actual population in year $5$ was $13.7$ thousand. What is the residual, in thousands, at $t=5$?

The function $g$ has the property that for each time the output values double, the input values increase by 1. Which of the following could be the graph of $y=g\left(x\right)$ in the $xy$-plane?

The increasing function $N$ gives the number of subscribers, in thousands, for a new streaming channel. The table gives values of $N\left(t\right)$ for selected values of $t$, in months, since the channel launched.

If a model is constructed to represent these data, which of the following best applies to this situation?

The table gives values for a function $p$ at selected values of $x$.

Which of the following graphs could represent these data in a semi-log plot, where the vertical axis is logarithmically scaled?

Two function models $f$ and $g$ are constructed to represent the temperature of a liquid being heated. Both $f\left(t\right)$ and $g\left(t\right)$ represent the temperature, in degrees Celsius, after $t$ minutes for $t\ge 1$. If $f\left(t\right)=80+12\ln t$ and $g\left(t\right)=3t+77$, what is the first time $t$ that the temperature predicted by the logarithmic model will be 0.5 degrees more than the temperature predicted by the linear model?

A biologist records the number of bacteria, in thousands, in a culture sample at the end of each day. The data are shown in the table.

The biologist uses exponential regression to fit a model of the form $B\left(d\right)=a{b}^d$ to the data. What is the value predicted by the model for the bacteria count, in thousands, when $d=2.5$?

The table gives values for a function $q$ at selected values of $x$.

Which of the following graphs could represent these data in a semi-log plot, where the vertical axis is logarithmically scaled?

A gym owner believes there is a linear relationship between monthly advertising spend and the number of new memberships signed up. After fitting a linear regression model, the residuals are plotted against advertising spend, as shown in the figure. Which of the following statements about the linear regression model is true?

Two function models $f$ and $g$ are constructed to represent the speed of a racing car. Both $f\left(t\right)$ and $g\left(t\right)$ represent the speed, in kilometres per hour, after $t$ minutes for $t\ge 1$. If $f\left(t\right)=60+8\ln t$ and $g\left(t\right)=2t+58$, what is the first time $t$ that the speed predicted by the logarithmic model will be 0.5 kilometres per hour more than the speed predicted by the linear model?

The number of subscribers, in thousands, of a streaming service is modelled by the function $S$. The number of subscribers is expected to increase by $4\%$ each month. At time $t=0$ years, the service has $200$ thousand subscribers. If $t$ is measured in years, which of the following is an expression for $S\left(t\right)$?

(Note: There are 12 months in a year.)

A radioactive substance has a half-life of $24$ minutes. In a particular sample, the amount of the substance remaining after $m$ minutes is modelled by $h\left(m\right)=A_0(0.5{)}^{m/24}$, where $A_0$ is the amount at time $m=0$. Which of the following functions $k$ models the amount of the substance remaining after $s$ seconds, where $A_0$ is the amount at time $s=0$?

(Note: There are 60 seconds in a minute, so $m=\frac{s}{60}$.)

A scientist measures the number of bacterial colonies in a culture, in thousands, at hourly intervals. The table gives the number of colonies $y$ at time $x$, where $x$ is the number of hours since the start of the experiment.

Which of the following functions best models the data?

An exponential model is constructed to predict the population of a species in a wildlife reserve. The reserve has a maximum sustainable population of 5,000 individuals (the carrying capacity). Initially the model fits the population well, but over a longer time horizon the model's predictions become increasingly inaccurate. Which of the following best explains why the model's error increases over time?

The number of registered users for a social media app, in millions, is modeled by the function $U$ given by $U\left(t\right)=2.5+4.1\ln \left(t+1\right)$, where $t$ is the number of months since the app launched. Based on the model, what is the value of $t$, as a decimal approximation, when the number of registered users first reaches $12$ million?

The temperature, in degrees Celsius, of a metal rod placed in an oven $t$ minutes after being placed is modeled by the function $T$ given by $T\left(t\right)=a+b\ln \left(t+1\right)$, where $a$ and $b$ are constants. The temperature of the rod at $t=0$ is $20$ degrees Celsius, and at $t=7$ minutes is $50$ degrees Celsius.

According to the model, what is the temperature of the rod at $t=15$ minutes?

In the $xy$-plane, the graph of ${\log }_{2}y$ as a function of $x$ is a line that passes through the points $\left(0,3\right)$ and $\left(2,7\right)$.

Which of the following is the exponential function $y=a·b^x$ that models the relationship between $x$ and $y$?

A function $f$ is modeled by $f\left(x\right)=a·b^x$, where $a$ and $b$ are constants. The graph of $f$ passes through the points $\left(2,12\right)$ and $\left(5,96\right)$.

According to the model, what is the value of $f\left(7\right)$?

The function $f$ is given by $f\left(x\right)=a·b^x$, where $a$ and $b$ are positive constants. If $f\left(1\right)=6$ and $f\left(4\right)=162$, what is the value of $a+b$?

The sound level $L$, in decibels, is related to sound intensity $I$ by the model $L=10{\log }_{10}\left(\frac{I}{I_0}\right)$, where $I_0$ is a reference intensity. If the sound level increases by $20$ decibels, by what factor does the intensity $I$ increase?

A bank account balance, in dollars, after $t$ years is modeled by $B\left(t\right)=5000 e^{0.04t}$. In how many years will the balance first equal $10,000?