Exponents & Logarithms (ln, log_10, integer exponents): notes and practice questions

The Richter scale measures the magnitude of an earthquake using the formula:

$M = \log_{10}\left( \frac{I}{I_{0}} \right)$

Where $M$ is the magnitude, $I$ is the intensity of the earthquake, and $I_{0}$ is a reference intensity.

$a$ Find the magnitude of an earthquake with an intensity $7.5 \times 10^{4}$ times greater than the reference intensity $I_{0}$.

(b.i) Write an expression for I in terms of $M$ and $I_{0}$

(b.ii) Find the intensity of an earthquake with magnitude 5.6, given the reference intensity $I_{0} = 10^{- 12}\ Wm^{- 2}.$ Give your answer in the form $b \times 10^{a}$, where $1 \leq b < 10$ and $a$ is an integer

(a) A water purification system reduces the impurity concentration in water. Initially, the impurity concentration is 320 mg/L. Each filter stage removes 15% of the impurities present before that stage.

Show that the impurity concentration after the fourth filter stage is approximately 167 mg/L, to the nearest mg/L.

(b) Find the minimum number of filter stages required for the impurity concentration to be less than 25 mg/L.

(c) A technician records the impurity concentration after each stage. Calculate the sum of the concentrations recorded after the first, second, third, and fourth stages. Give your answer to one decimal place.

A manufacturing company purchases a new specialized machine. The machine's value depreciates exponentially over time. The value of the machine, $V$, in thousands of dollars, $t$ years after its purchase, is modelled by the function $V(t) = A e^{-kt}$, for $t \ge 0$.

The initial value of the machine was 150 thousand dollars. After 3 years, its value had decreased by 40%.

(a) Find the value of $k$.

(b) Calculate the value of the machine after 5 years and 6 months, giving your answer to two decimal places.

(c) The company believes that, according to this model, the machine will always have some residual value, however small.

State a mathematical reason why the company might believe this.

(d) Write down one possible limitation of the domain of the model.

(a) A scientist is modeling the growth rate of a specific enzyme in a culture. The instantaneous rate of change is given by the expression:

$\sqrt{12x^5} \times 5x^0 \times 2x^3$

Simplify this expression, assuming $x > 0$.

(b) An astrophysicist is analyzing the energy density fluctuations in a nebula. These fluctuations are described by a ratio of terms involving a variable $x$. Simplify the following expression:

$\frac{(x^{-3})^2}{(x^4)^{-1}}$

Express your answer with a positive exponent.

(c) An engineer is designing a new composite material, and the stress distribution in a critical section is modeled by the following expression involving variables $x$ and $y$:

$\frac{(8x^6)^{\frac{2}{3}}}{16y^8}$

Simplify this expression.

A small business owner, Ms. Chen, invests 8000 CHF into a long-term growth fund for her company's expansion. The fund offers a nominal annual interest rate of 3.2%, compounded monthly.

Calculate the total amount Ms. Chen will have in her fund after 7 years. Give your answer correct to 2 decimal places.

Another entrepreneur, Mr. Davies, wants to invest 12000 EUR such that his investment will grow to 1.75 times its initial amount in 10 years. Assume the investment account pays a nominal annual interest of $r$% compounded quarterly.

Determine the value of $r$.

(a) A population of bacteria in a petri dish grows according to a geometric sequence. On day 2, there are $12$ bacteria. On day 5, there are $324$ bacteria.

Find the initial number of bacteria (first term, $a$) and the daily growth factor (common ratio, $r$).

(b) Find:

(i) the number of bacteria on day 8.

(ii) the total number of bacteria observed from day 1 to day 8.

(c) Find the first day $n$ when the number of bacteria exceeds $500 \, 000$.

A biologist is studying the growth of a bacterial colony in a petri dish. The population of the colony, $N$, can be modelled by an exponential function

$N(t) = Ae^{kt}$

where $t$ is the time in hours since the start of the experiment, and $A$ and $k$ are constants.

At the start of the experiment, the bacterial colony has a population of 250 cells. After 4 hours, the population has grown to 800 cells.

Write down the value of $A$.

Find the value of $k$.

A landscape architect is designing a large decorative fountain for a new public park. The outer structure of the fountain consists of a cylindrical base topped by a conical section. The inner part of the fountain, which holds the water, is a hollow space created by rotating a specific curve around the vertical axis.

The shape of the inner hollow is based on a transformation of the graph $y = -x^3$. The curve that defines the profile of the inner hollow is given by $y = 40 - 0.064x^3$. This transformation involves a vertical translation of $a$ units and a stretch parallel to the x-axis with a scale factor of $b$.

(a.i) Write down the value of $a$.

(a.ii) Find the value of $b$.

The cylindrical base of the fountain has a radius of $15$ m and a height of $5$ m. The conical section on top has the same base radius of $15$ m and a height of $8$ m. The inner hollow, described by the curve $y = 40 - 0.064x^3$, extends from the base of the fountain ($y=0$) up to the total height of the outer structure.

Find the volume of the solid material that makes up the fountain (i.e., the volume of the outer structure minus the volume of the inner hollow).

A biologist is studying the effect of a new toxin on a bacterial culture. The number of active bacterial cells, $N$ (in thousands), remaining after exposure to a toxin dose $D$ (in mg/L) follows the relationship:

$\log_{10}N = k - D$, for some constant $k \in \mathbb{R}$.

In an experiment, it was observed that when the toxin dose was 2 mg/L, there were 1000 thousand (i.e., 1,000,000) active cells remaining.

(a) Find the value of $k$.

The relationship for this bacterial culture can also be written in the form $N = \frac{b}{10^D}$.

(b) Find the value of $b$.

(c) Given that the toxin dose $D$ is between 0.5 mg/L and 4.0 mg/L (i.e., $0.5 < D < 4.0$ ), find the range for $N$.

The effectiveness score, $S$, of a new antidote is inversely proportional to the number of active cells, $N$, remaining after toxin exposure, such that $S = \frac{1}{N}$.

(d) A new antidote was tested on a culture exposed to a dose of 4.5 mg/L. Calculate the effectiveness score, $S$, for this antidote. Give your answer to 3 significant figures.

A tech company launches a new social media app. The number of active users, $U$, can be modelled by the function

$U(t) = 800 \times k^t, t\ge 0$,

where $t$ is the number of hours since the app was launched, and $k$ is a positive constant.

Write down the value of $U(0)$.

Interpret what this value means in this context.

4 hours after the app was launched, the number of active users was 4050.

Find the value of $k$.

Find the number of active users 2 hours and 15 minutes after the app was launched.

A competitor launches a similar app, whose user base, $U_2$, can be modelled by the function

$U_2(t) = 2500 \times 1.2^t, t\ge 0$,

where $t$ is the number of hours since both apps were launched.

Find the value of $t$ when the number of users for both apps is equal.

It takes $H$ hours and $m$ minutes for the number of users of the first app to reach 15000.

Find the value of $H$ and $m$, giving $m$ as an integer.

Each user of the first app requires $1.5 \times 10^{-2}$ MB of server storage. The total available server capacity is $3.0 \times 10^5$ MB.

Determine how long it would take for the app's user base to exceed the server capacity.

A new experimental drug is administered to a patient. The concentration of the drug in the patient's bloodstream, $C$, in mg/L, $t$ hours after administration, is modelled by the function $C(t) = C_0 e^{-kt}$, where $C_0$ and $k$ are positive constants.

Initially, the concentration of the drug is $250$ mg/L. After $2$ hours, the concentration drops to $150$ mg/L.

Determine the value of $k$.

Using this model, calculate the concentration of the drug in the bloodstream $5$ hours after administration.

Based on this model, will the drug ever completely leave the patient's bloodstream? Justify your answer.

State one limitation of the domain of this model in a real-world context.

A botanical garden is studying the growth of a rare plant species, Species A.

The population of Species A, $N_A$, can be modelled by the function

$N_A(t) = 800 \times k^t, t \ge 0$,

where $t$ is the number of years since the study began, and $k$ is a positive constant.

Write down the value of $N_A(0)$.

Interpret what this value means in this context.

4 years after the study began, the population of Species A is 4050.

Find the value of $k$.

Find the population of Species A 2 years and 6 months after the study began.

The botanical garden also studies a second rare plant species, Species B.

The population of Species B, $N_B$, can be modelled by the function

$N_B(t) = 2500 \times 1.2^t, t \ge 0$,

where $t$ is the number of years since both studies began.

Find the value of $t$ when the populations of the two plant species are equal.

It takes $Y$ years and $M$ months for the number of plants in Species B to reach 10000.

Find the value of $M$, giving your answer as an integer value.

A colony of bacteria is growing in a nutrient solution. The rate of change of the population, $P$, with respect to time, $t$ (in hours), is modelled by the differential equation $\frac{dP}{dt} = P \cos t (e^{-\sin t})$. At time $t = 0$, the population is $P = 10$.

(a) By using Euler's method with a step length of 0.1, find an approximate value for the population when $t = 0.3$. Give your answer to three significant figures.

(b) By solving the differential equation, find the percentage error in your approximation for the population when $t = 0.3$. Give your answer to three significant figures.

(a) A new social media app, "Connectify", reported a user base of $15.73$ million users at the end of its first year.

(i) Write down $15.73$ million correct to the nearest million.

(ii) Find the percentage error if $15.73$ million is rounded to the nearest million.

Connectify's initial user growth followed an arithmetic progression. In the first month, they gained $5000$ new users. In the second month, they gained $7000$ new users, and in the third month, $9000$ new users.

(b) Find the month during which Connectify gained $35000$ new users.

(c) Calculate the total number of new users Connectify gained in the first $12$ months.

Meanwhile, LinkUp, a rival platform, started with an advertising budget of $100000$ in the first quarter. Due to its success, they decided to increase their budget by $15\%$ each subsequent quarter.

(d) Determine the first quarter in which LinkUp's advertising budget exceeds $500000.

(e) Find the first quarter that the total advertising expenditure by LinkUp, since the start, exceeds $2000000.

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 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 new online challenge is introduced. In the first hour, 120 people participate. Due to decreasing novelty, the number of new participants joining each subsequent hour is 82% of the number of new participants who joined the previous hour.

(a) Show that the number of new participants joining in the 7th hour is 36, to the nearest whole number.

(b) Find the number of hours for which the number of new participants joining in that hour is greater than 15.

(c) Find the total number of people who have heard about the challenge by the end of the 5th hour, to the nearest whole number.

This question explores models for the temperature of a cooling metal object.

A metal object is heated and then allowed to cool in a room where the ambient temperature is $20$ °C. The temperature, $T$ °C, of the object is recorded every $5$ minutes, starting from $t = 0$ minutes.

The data is first modelled using a linear function, $T(t) = at + b$, where $a, b \in \mathbb{R}$.

Find the equation of the regression line of $T$ on $t$.

Interpret the meaning of the parameter $a$ in the context of the model.

Suggest why using this linear regression equation to predict the time it will take for the object to cool to $20$ °C could be unreliable.

The data is then modelled using a quadratic function, $T(t) = pt^2 + qt + r$, where $p, q, r \in \mathbb{R}$.

Find the equation of the least squares quadratic regression curve.

Use this quadratic equation to predict the time it will take for the object to cool to $40$ °C.

Hence, write down a suitable domain for the function $T(t) = pt^2 + qt + r$ in the context of this cooling process.

A metal object with a constant heat capacity $C$ Joules/°C is cooling in a room. The rate of heat transfer from the object to its surroundings is given by $P = \alpha A (T - T_a)$, where $P$ is the rate of heat loss in Joules/minute, $\alpha$ is the heat transfer coefficient, $A$ is the surface area of the object, $T$ is the object's temperature, and $T_a$ is the ambient temperature. The rate of change of the object's internal energy is given by $\frac{dE}{dt} = C \frac{dT}{dt}$.

Given that the ambient temperature is $T_a = 20$ °C and assuming that all heat loss is due to this process, show that the differential equation for the temperature of the object can be written as $\frac{dT}{dt} = -K(T - 20)$, where $K$ is a positive constant.

By solving the differential equation $\frac{dT}{dt} = -K(T - 20)$, show that the general solution is given by $T = 20 + Ae^{-Kt}$, where $A \in \mathbb{R}$.

Use the general solution from part (d) and the initial condition $T(0) = 90$ °C, along with the data point $T(5) = 74.5$ °C, to find the values of $A$ and $K$. Hence, predict the time it takes for the object to cool to $25$ °C.

If the object was initially heated to a different temperature, $T_0'$ °C, such that it cools to $30$ °C in $30$ minutes, find this new initial temperature $T_0'$. Use the value of $K \approx 0.0500589$ from part (e).

Now, consider a scenario where the object is cooling, but also receives a small amount of heat from an external source that decreases over time. The temperature of the object, $T$ °C, is modelled by the differential equation $\frac{dT}{dt} = -0.05(T - 20) + 1.5 e^{-0.1t}$.

Given the initial temperature $T(0) = 90$ °C, use Euler's method with a step length of $h = 2$ minutes to estimate the temperature of the object at $t = 6$ minutes.

The pH of a solution measures its acidity and can be determined using the formula

$\text{pH} = -\log_{10} C$, where $C$ is the concentration of hydronium ions in the solution, measured in moles per litre. A lower pH indicates a more acidic solution.

The concentration of hydronium ions in a particular brand of orange juice is $2.5 \times 10^{-4}$ moles per litre.

(a) Calculate the pH of this orange juice.

(b) A different type of juice, lemon juice, has 8 times the concentration of hydronium ions of the orange juice in part (a).

Determine whether the lemon juice is more or less acidic than the orange juice.

Justify your answer mathematically.