Exponents: notes and practice questions

(a) Find the value of $\log_4 64$.

(b) Find the value of $\log_3 \left(\frac{1}{9}\right)$.

(c) Find the value of $\log_{25} 5$.

(a) A scientist is studying the rate at which a certain chemical compound dissolves in a solution. The rate of dissolution, $R$, in grams per minute, at time $t$ minutes, is modeled by the expression $R(t) = \frac{4\sqrt{t} - 7}{\sqrt{t}}$.

This expression can be written in the form $A - Bt^p$, where $A$, $B$, and $p$ are constants. Write down the value of $p$.

(b) Hence, calculate the total amount of compound dissolved between $t=4$ minutes and $t=9$ minutes.

In a study of wave propagation, a mathematical model uses the function $g(x) = \frac{e^x - e^{-x}}{2}$, where $x \in \mathbb{R}$, to describe a certain physical quantity. This function is also known as the hyperbolic sine function, $\sinh x$.

Verify that $y = g(x)$ satisfies the differential equation $\frac{d^2y}{dx^2} = y$.

Another related function, the hyperbolic cosine, is defined as $f(x) = \frac{e^x + e^{-x}}{2}$, also known as $\cosh x$. Show that $(\cosh x)^2 - (\sinh x)^2 = 1$.

The functions $\cosh x$ and $\sinh x$ can be extended to complex numbers. Using Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$, where $\theta \in \mathbb{R}$, express $\cosh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Similarly, express $\sinh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Hence, show that $(\cosh(i\theta) )^2 + (\sinh(i\theta) )^2 = \cos(2\theta)$.

In a design project, a component's profile is described by a hyperbola with parametric equations $x = A \cosh t$ and $y = B \sinh t$, where $A, B$ are positive constants and $t \in \mathbb{R}$.

Given that the component's profile passes through the point $(6, 0)$ and has asymptotes $y = \pm \frac{4}{3}x$, find the values of $A$ and $B$.

A biologist is modelling the growth of two different bacterial colonies. The first colony, A, grows such that its population at time $x$ is given by $P_A(x) = a^x$, where $a$ is a growth factor and $x \ge 0$. The second colony, B, grows linearly such that its population at time $x$ is $P_B(x) = x$.

Consider the cases where the growth factor $a = 2$ and $a = 10$. On the same set of axes, sketch the following three graphs for $x \ge 0$:

$y = 2^x$

$y = 10^x$

$y = x$

Clearly label each graph with its equation and state the coordinates of any non-zero $y$-axis intercepts.

In parts (b) and (c), consider the case where the growth factor $a = e$.

Use calculus to find the minimum value of the expression $e^x - x$, justifying that this value is a minimum.

Hence deduce that $e^x > x$ for all $x \in \mathbb{R}$.

There exist values of $a$ for which the graph of $y = a^x$ and the line $y = x$ have different numbers of intersection points. The following table gives three intervals for the value of $a$.

By investigating the graph of $y = a^x$ for different values of $a$, write down the values of $p, q$ and $r$.

In parts (e) and (f), consider $a \in \mathbb{R}^+, a \neq 1$.

For $1.4 \leq a \leq 1.5$, a value of $a$ exists such that the line $y = x$ is a tangent to the graph of $y = a^x$ at a point P.

Find the exact coordinates of P and the exact value of $a$.

Write down the exact set of values for $a$ such that the graphs of $y = a^x$ and $y = x$ have

(i) two intersection points;

(ii) no intersection points.

Show that $(\sqrt{x} + \frac{1}{\sqrt{x}})^2$ can be written as $x + 2 + \frac{1}{x}$.

Hence, find the exact value of $\int_{1}^{e^2} (\sqrt{x} + \frac{1}{\sqrt{x}})^2 dx$.

Consider the function $f(x) = a^x$ where $x, a \in \mathbb{R}$ and $a > 1$. The graph of $f$ contains the point $(\frac{3}{2}, 27)$.

(a) Show that $a = 9$.

(b) Write down an expression for $f^{-1}(x)$.

(c) Find the value of $f^{-1}(\frac{1}{81})$.

Consider the arithmetic sequence $\log_9 8, \log_9 p, \log_9 q, \log_9 27$, where $p > 1$ and $q > 1$.

(i) Show that $8, p, q$ and $27$ are four consecutive terms in a geometric sequence.

Consider the arithmetic sequence $\log_9 8, \log_9 p, \log_9 q, \log_9 27$, where $p > 1$ and $q > 1$.

(ii) Find the value of $p$ and the value of $q$.

The expression $\frac{2\sqrt[3]{x}-5}{\sqrt[3]{x}}$ can be written as $2-5x^{p}$. Write down the value of $p$.

Hence, find the value of $\int_{1}^{8} \left(\frac{2\sqrt[3]{x}-5}{\sqrt[3]{x}}\right) dx$.

The rate of change of a certain quantity $R$ with respect to a variable $x$ is given by $R'(x)=\frac{1}{x(M-x)}$, $x \in \mathbb{R}$, $x \neq 0$, $x \neq M$ where $M$ is a positive constant.

The expression for $R'(x)$ can be written in the form $\frac{A}{x} + \frac{B}{M-x}$, where $A, B \in \mathbb{R}$.

Find $A$ and $B$ in terms of $M$.

Hence, find an expression for $R(x)$.

The concentration of a certain chemical product, $C$ (in mol/L), in a reaction vessel at time $t$ (in minutes) can be modelled by the differential equation $\frac{dC}{dt} = \frac{C(L-C)}{8L}$, where $L$ is the maximum possible concentration and $C(0) = 0.2$ mol/L is the initial concentration.

By solving the differential equation, show that $C = \frac{0.2 L}{(L-0.2)e^{-\frac{t}{8}}+0.2}$.

At $t=12$ minutes, the concentration of the product has reached $0.6$ mol/L.

Find the value of $L$, giving your answer correct to four significant figures.

Find the value of $t$ when the rate of change of the concentration is at its maximum.

The following table shows the annual revenue of a tech startup, Quantum Innovations, $t$ years after its launch in 2015.

A data analyst uses linear regression to model the revenue of Quantum Innovations using these data.

The analyst's model is $R = at + b$.

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

(a)(ii) Interpret, in context, the value of $a$.

(b) The analyst uses this model to predict the revenue of Quantum Innovations in the year 2030, where $t = 15$, and calculates a revenue of approximately $11.8$ million USD.

Comment on the reliability of the analyst's prediction.

(c)(i) A financial expert, Elena, develops an exponential model for Quantum Innovations' future revenue.

In this model, $R_E(t) = 1.6(1.08)^t$ represents the revenue in millions of USD $t$ years after 2015, where $10 \le t \le 25$.

Use Elena's model to predict the revenue of Quantum Innovations in the year 2035.

(c)(ii) Interpret, in context, the value $1.08$ in Elena's model.

(d) Another financial expert, Carlos, develops a third model for Quantum Innovations' revenue.

In this model, $R_L(t) = \frac{20}{1+21e^{-0.15t}}$ represents the revenue in millions of USD $t$ years after 2015, where $10 \le t \le 25$.

Use Carlos's model to predict the revenue of Quantum Innovations in the year 2035.

(e) Determine the year in which the difference between the predictions from Elena's model and Carlos's model is greatest.

(f)(i) Find the value of

$R_E'(18)$;

(f)(ii) Find the value of

$R_L'(18)$.

(g) Compare and interpret, in context, the values of $R_E'(18)$ and $R_L'(18)$.

A scientist is studying a spherical microbe. The radius of the microbe is measured to be $3 \times 10^{-5}$ cm.

(a) Write down the diameter of the microbe.

(b) The volume of the microbe can be expressed in the form $\pi(a \times 10^k) \text{ cm}^3$ where $1 \leq a < 10$ and $k \in \mathbb{Z}$.

Find the value of $a$ and the value of $k$.

A company is designing a new power generator. The power output, $P$, in megawatts (MW), of a prototype generator at time $t$ hours after startup is modelled by the function $P(t) = t^3 - 3ct + K$, where $t \in \mathbb{R}$, $c$ is a control parameter, and $K$ is a constant representing the initial power. For parts (a) to (e), assume $K = 4$.

On separate axes, sketch the graph of $P = P(t)$ showing the value of the $P$-intercept and the coordinates of any points with zero gradient, for

(i) $c = 1$;

(ii) $c = 2$.

Write down an expression for $P'(t)$.

Hence, or otherwise, find the set of values of $c$ such that the graph of $P = P(t)$ has

(i) a point of inflexion with zero gradient;

(ii) one local maximum point and one local minimum point;

(iii) no points where the gradient is equal to zero.

Given that the graph of $P = P(t)$ has one local maximum point and one local minimum point, show that

(i) the $P$-coordinate of the local maximum point is $2c^{\frac{3}{2}} + 4$;

(ii) the $P$-coordinate of the local minimum point is $-2c^{\frac{3}{2}} + 4$.

Hence, for $c > 0$, find the set of values of $c$ such that the graph of $P = P(t)$ has

(i) exactly one $t$-axis intercept;

(ii) exactly two $t$-axis intercepts;

(iii) exactly three $t$-axis intercepts.

Consider a modified power output function $Q(t) = t^3 - 3ct + d$ for $t \in \mathbb{R}$ and where $c, d \in \mathbb{R}$. Find all conditions on $c$ and $d$ such that the graph of $P = Q(t)$ has exactly one $t$-axis intercept, explaining your reasoning.

The expression $\frac{5x-2}{\sqrt[3]{x}}$ can be written in the form $5x^p - 2x^q$. Write down the value of $p$ and the value of $q$.

Hence, find the value of $\int_1^8 \frac{5x-2}{\sqrt[3]{x}} \, dx$.

A pharmaceutical company is formulating a new drug. They are testing two active ingredients, $x_1$ and $x_2$, such that their total concentration is $24$ mg/mL. The drug's efficacy is modelled by the product of the concentrations of the two ingredients.

Find the efficacy, $E$, as a function of $x_1$ only.

Determine the concentration of $x_1$ that maximizes the drug's efficacy.

Hence, show that the maximum efficacy for two ingredients with a total concentration of $24$ mg/mL is $144$.

Let $M_n(S)$ represent the maximum efficacy for a drug with $n$ active ingredients and a total concentration of $S$ mg/mL. For $n = 2$, the maximum efficacy can be expressed as $M_2(S) = \left(\frac{S}{2}\right)^2$.

Verify that $M_2(S) = \left(\frac{S}{2}\right)^2$ is true for $S = 24$.

The relationship between the geometric mean and arithmetic mean states that for $n$ positive real numbers $x_1, x_2, ..., x_n$, their geometric mean $(x_1 \times x_2 \times ... \times x_n)^{\frac{1}{n}}$ is always less than or equal to their arithmetic mean $\frac{x_1 + x_2 + ... + x_n}{n}$.

Show that the geometric mean and arithmetic mean are equal when $x_1 = x_2 = ... = x_n$.

Use this result to prove that $M_n(S) = \left(\frac{S}{n}\right)^n$.

Using the formula for $M_n(S)$, determine the value of:

$M_3(24)$;

$M_4(24)$;

$M_5(24)$.

For a fixed total concentration of $S = 24$ mg/mL, the company wants to find the optimal number of active ingredients, $n$, to maximize the drug's efficacy. Let $P(S)$ denote this maximum efficacy.

Write down the value of $P(24)$ and the value of $n$ at which it occurs.

Determine the value of $P(30)$ and the value of $n$ at which it occurs.

Consider the continuous function $h$, defined by $\ln(h(x) ) = x\ln\left(\frac{S}{x}\right)$, where $x \in \mathbb{R}^+$. A sketch of the graph of $y = h(x)$ is shown in the following diagram. Point A is the maximum point on this graph.

Find, in terms of $S$, the $x$-coordinate of point A.

Verify that $h(x) = M_x(S)$, when $x \in \mathbb{Z}^+$.

The company has a total concentration of $S = 150$ mg/mL available. Use your answer to part (h) to find the largest possible efficacy. Give your answer in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}^+$.

Consider the series $e^{2x} + k e^{2x} + \frac{1}{5}e^{2x} + ...$, where $x \in \mathbb{R}$ and $k \in \mathbb{R}, k \neq 0$.

(a) Consider the case where the series is geometric.

(i) Show that $k = \pm \frac{1}{\sqrt{5}}$.

(ii) Given that $k > 0$ and the sum to infinity is $\frac{5+\sqrt{5}}{4}$, find the value of $x$.

(b) Now consider the case where the series is arithmetic.

(i) Show that $k = \frac{3}{5}$.

(ii) Write down the common difference, $d$, in the form $c e^{2x}$ where $c \in \mathbb{Q}$.

(iii) The sum of the first $n$ terms of the series is $-8e^{2x}$. Find the value of $n$.

Consider the complex number $z = 2^{1-i}$.

(a) Write the integer 2 in the form $e^k$ where $k \in \mathbb{R}$.

(b) Hence, find $z$ in the form $x+iy$, where $x$ and $y$ are expressed in terms of $\ln 2$.

(c) Find $\text{Im}(z + z^{-1})$, where $z^{-1}$ is the multiplicative inverse of $z$.