Laws of logarithms (adding/product, subtracting/quotient, distribution): notes and practice questions

A mathematical model for a physical phenomenon involves the expression $\left(\frac{1}{u} + 2\right)^2$.

(a) Expand $\left(\frac{1}{u} + 2\right)^2$.

The rate of change of a certain quantity is given by $f'(x) = \frac{1}{(x+1)^2} + \frac{4}{x+1} + 4$.

(b) Find the indefinite integral $\int \left( \frac{1}{(x+1)^2} + \frac{4}{x+1} + 4 \right) dx$.

A designer is creating a custom-shaped container. The cross-sectional profile of the container is defined by the curve $y = \frac{1}{x+1} + 2$ for $1 \le x \le 3$. The container is formed by rotating this region $2\pi$ about the x-axis.

(c) Calculate the volume of the solid formed. Give your answer in the form $\frac{\pi}{4}(a + b \ln(c) )$, where $a, b, c \in \mathbb{Z}$.

A car enthusiast, Mr. Henderson, purchases two classic cars for his collection. The first is a popular sedan, initially costing $28000, which is expected to depreciate at a rate of 8$75000, and is expected to depreciate at a rate of 15% per year.

(a) Estimate the value of the popular sedan after 4 years.

(b) Find the number of years, $k$, after which both cars will have the same estimated value. Give your answer to three significant figures.

(c) Comment on the validity of your answer to part (b).

(a) A tech startup's monthly revenue is modelled as a geometric sequence. The revenue in the first month was $R_1$ dollars, and it increases by a constant factor $k$ each month.

(i) Write down an expression in terms of $R_1$ and $k$ for the revenue in the $n$th month, $R_n$.

(ii) Write down an expression for the total revenue generated over the first $N$ months, $S_N$.

(b) The marketing team measures the 'growth potential' $G_n$ of the startup in the $n$th month using the formula $G_n = \log_{10} R_n$.

By writing $G_n$ in terms of $R_1$ and $k$, determine what type of sequence $G_n$ defines, and find an expression for the $n$th term $G_n$. You must clearly show your working and justify your answer.

(c) Find an expression for the sum $T_N$ of the growth potential over the first $N$ months.

(d) Determine whether or not $T_N = \log_{10} S_N$. Justify your answer.

(a) A chemical's toxicity score, $T$, is modelled as a function of its relative concentration, $C$, by the equation:

$T = 5 - 3 \log_{10}(C)$

where $C$ is the concentration relative to a baseline. A new compound, 'Zylos', is being tested and has a relative concentration of $0.085$.

Find the Toxicity Score of Zylos.

(b) Another chemical compound, 'Hydron', is found to have a Toxicity Score of 11.

Find the relative concentration of Hydron.

(c) Compound 'Alpha' has a Toxicity Score that is 4.5 units lower than Compound 'Beta'.

Find how many times greater the concentration of 'Alpha' is compared to 'Beta'.

(a) A manufacturing company uses a specialized machine to produce custom parts. The number of units produced, $N$, in one day is modelled by

$N = 40 \ln(3t+1)$, for $0 \le t \le 10$

where $t$ is the time the machine operates, in hours, on that day.

Find the time, in hours, it takes for the machine to produce 80 units in one day.

(b) The daily profit, $P$, in thousands of dollars, from producing $N$ units is given by

$P = 150(1-e^{-0.008N})$, for $P \in \mathbb{R}$.

Find the profit, in thousands of dollars, the company earns for producing 80 units.

(c) Find an expression for $P$ as a function of $t$, giving your answer in the form

$P=150(1-\frac{1}{(3t+1)^n})$

where $n$ is a number to be determined.

(d) Hence or otherwise, find the profit, in thousands of dollars, the company earns by operating the machine for 4 hours.

(e) Find the greatest profit, in thousands of dollars, the company can earn in one day.

(f) The company introduces a weekly bonus profit, $B$, in thousands of dollars, for the total units produced in one week, $D_{total}$, using

$B=100(1-\frac{1}{D_{total}+1})$

The company decides to operate the machine for the same length of time, $T$ hours, each day for 5 days a week.

They want to achieve a total weekly profit of 580 thousand dollars.

Find the value of $T$.

(a) The growth of a bacterial colony's area, $A$, over time, $t$, is being studied under specific conditions. The area $A$ is measured in square micrometers ($\mu m^2$) at the end of each hour $t$ for five consecutive hours. The data is recorded in Table 1.

**Table 1: Time $t$ (hours) and Area $A$ ($\mu m^2$)**

It is believed that the growth of the colony can be modelled by an equation of the form $A = k \times t^p$.

Use power regression on your graphic display calculator to find the value of $k$ and the value of $p$. Give your answers to three significant figures.

(b) The values of $t$ and $A$ can be transformed such that $x = \ln t$ and $y = \ln A$. Table 2 shows data for $x$ and $y$ to three decimal places.

**Table 2: Transformed data $x = \ln t$ and $y = \ln A$**

Find the linear regression equation of $y$ on $x$, in the form $y = cx + d$. Give the values of $c$ and $d$ to three decimal places.

(c) Hence, show that this linear regression is equivalent to the power regression found in part (a).

In this question, marine biologists are studying the population dynamics of a rare species of "Azurefin" fish in a newly established protected reef area.

Historically, the Azurefin population in this region maintained a stable size of $5000$ individuals. Following a period of environmental disturbance, the population was reduced to $1000$ fish. At this point, the area was designated a protected marine reserve, and conservation efforts began, leading to a recovery in the fish population.

Researchers wish to model the size of the Azurefin population, $x$, as a function of $t$, where $t$ is the time, in years, since the establishment of the protected reserve.

Initially, the researchers consider using the logistic model:

$x = \frac{L}{1+Ce^{-kt}}$, where $L, C, k \in \mathbb{R}^+$.

The researchers decide to set $L = 5000$.

State the assumption being made by setting $L = 5000$.

At $t = 0$, the population of Azurefin fish is $1000$.

Find the value of $C$.

At $t = 3$ years, the population of Azurefin fish is found to have increased to $2500$.

Find the value of $k$. Give your answer correct to three significant figures.

Use your model to predict the size of the Azurefin population in the area $6$ years after it became protected. Give your answer correct to the nearest whole number.

An alternative model for population growth is called the Gompertz model. When applied by the researchers to the Azurefin population, this model satisfies the differential equation:

$\frac{dx}{dt} = ax \ln \left( \frac{5000}{x} \right)$, $a \in \mathbb{R}^+$.

Write down the value of $\frac{dx}{dt}$ when $x = 5000$.

Interpret your answer to part (b)(i) in context.

Consider the function $f(x) = \ln (\ln 5000 - \ln x)$, where $0 < x < 5000$.

Show that $f'(x) = \frac{-1}{x \ln \left( \frac{5000}{x} \right)}$.

Hence, use separation of variables to show that the general solution of

$\frac{dx}{dt} = ax \ln \left( \frac{5000}{x} \right)$, where $0 < x < 5000$,

can be written as

$\ln x = \ln 5000 - Ae^{-at}$,

where $A$ is an arbitrary positive constant.

Use the size of the Azurefin population at $t = 0$ to find the value of $A$.

Give your answer in the form $A = \ln p$, where $p \in \mathbb{Z}^+$.

Use the size of the Azurefin population at $t = 3$, given in part (a), to show that $a = 0.281$, correct to three significant figures.

Use the Gompertz model to predict the size of the Azurefin population at $t = 6$. Give your answer correct to the nearest whole number.

After $6$ years, the Azurefin population is measured and is found to be $3200$.

Comment on the predictions made by the two models.

By tracking individual Azurefin fish, the researchers find that about $5\%$ of the population migrates out of the protected area each year.

They decide to adapt the Gompertz model to allow for this. The new model will satisfy the differential equation:

$\frac{dx}{dt} = 0.280799x \ln \left( \frac{5000}{x} \right) - 0.05x$.

Use Euler's method, with a step size of $0.5$ years and an initial value of $x_0 = 2500$ when $t = 3$, to find an estimate for the size of the Azurefin population when $t = 6$.

Give your answer correct to the nearest whole number.

Comment on your answer.

The rate of change of a certain quantity, $Q$, with respect to time, $t$ (in minutes), is modelled by the function $\frac{dQ}{dt} = \frac{10}{5t-2}$.

(a) Find an expression for $Q(t)$ in terms of $t$, assuming an arbitrary constant of integration.

(b) Given that the model is valid for $t > \frac{2}{5}$, find the exact total change in the quantity $Q$ from $t=1$ minute to $t=4$ minutes. Give your answer in the form $a \ln b$, where $a, b \in \mathbb{N}$.

In a microbiology experiment, the growth of two different bacterial cultures, Culture A and Culture B, is monitored. The logarithm (base 10) of their current populations, $P_A$ and $P_B$ respectively, are defined as $\log_{10} P_A = m$ and $\log_{10} P_B = n$. Express the following logarithmic expressions in terms of $m$ and $n$.

$\log_{10} (P_A P_B)$

$\log_{10} \left(\frac{P_A}{P_B}\right)$

$\log_{10} (P_A^4)$

$\log_{10} (P_A^2 P_B^3)$

$\log_{10} \left(\frac{\sqrt{P_A}}{P_B^5}\right)$

$\log_{10} (0.1 P_A^3)$

(a) A medical isotope used in diagnostic imaging decays over time. The time $t$ (in hours) since the isotope was prepared can be modelled by the equation $t = \frac{-12}{\ln 2} \ln(P)$, where $P$ is the proportion of the isotope remaining.

(i) Calculate the time $t$ (in hours) when the proportion of the isotope remaining is $P = 0.5$.

(ii) Calculate the time $t$ (in hours) when the proportion of the isotope remaining is $P = 0.25$.

(b) The isotope is considered ineffective if the proportion remaining drops below $0.05$. Find the time (in hours) when the isotope just becomes ineffective, giving your answer to one decimal place.

(c) Express $P$ in terms of $t$.

(d) Using your answer from part (c), find the proportion of the isotope remaining after $24$ hours. Give your answer to three significant figures.

The concentration, $C(t)$ in parts per million (ppm), of a pollutant in a lake after a clean-up operation can be modelled by the equation $C(t) = 5 + 75 \times (1.05)^{-t}$, where $t$ is the time in days since the operation began.

(a) Calculate the initial concentration of the pollutant in the lake.

(b) Determine the concentration of the pollutant after $20$ days.

(c) Find the number of days it takes for the pollutant concentration to reach $25$ ppm.

(d) State the long-term concentration of the pollutant in the lake.

A school administrator is tracking the spread of a rumour among students. The number of students, $S$, who have heard the rumour is recorded $t$ hours after it began. The first results obtained are shown in the top two rows of the table below.

Complete the last row of the table, giving your answers to three decimal places.

Draw a graph of $\log_{10} S$ against $t$, using appropriate scales on the axes.

Hence, state the type of model that best fits the data displayed in part (b).

The school counsellor suggests that the number of students who have heard the rumour can be modelled by $S(t) = A \times B^t$.

Explain why the school counsellor is correct.

Hence, determine the values of the parameters $A$ and $B$ of the school counsellor's model, showing all your calculations. Give your answers to three significant figures.

(a) A pharmaceutical company is investigating the effectiveness of a new drug. The effectiveness, $S$, is modelled as a function of the concentration of an activator, $A$, present in the system by the equation $S = S_{max} e^{-\frac{C}{A}}$, where $S_{max}$ and $C$ are positive constants, and $A > 0$.

Show that $\frac{dS}{dA}$ is always positive.

(b) Given that $\lim_{A \to \infty} S = S_{max}$ and $\lim_{A \to 0} S = 0$, sketch the graph of $S$ against $A$.

(c) The model predicts that the graph of $\ln S$ against $\frac{1}{A}$ is a straight line.

(i) Write down the gradient of this line in terms of $C$.

(ii) Write down the y-intercept of this line in terms of $S_{max}$.

(d) The following data are collected from experiments, where $A$ is measured in $\text{mol dm}^{-3}$ and $S$ is measured in arbitrary units:

Find the equation of the regression line for $\ln S$ on $\frac{1}{A}$.

(e) Find an estimate of

(i) $C$;

(ii) $S_{max}$.

It is not required to state units for these values.

A cup of hot tea is placed in a room with a constant ambient temperature of 22°C. The temperature of the tea, $T$ in degrees Celsius, is recorded at various times, $t$ in minutes.

It is assumed that the temperature difference, $D = T - 22$, follows an exponential decay model. To linearize the data, a graph of $\text{ln}(D)$ is plotted against $t$. This graph is a straight line that passes through the points $(4, 3.8)$ and $(12, 2.2)$.

(a) Find the equation of the straight line in the form $\text{ln}(D) = mt + c$, where $m$ and $c$ are constants.

(b) Hence,

(i) find an expression for $D$ in terms of $t$, writing your answer in the form $D = Ae^{kt}$.

(ii) calculate the temperature of the tea, $T$, when $t = 20$ minutes.