Logarithms: 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 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.

Exponential relationships appear frequently in models of population growth and radioactive decay. It is often useful to express these relationships in logarithmic form.

(a) Write the equation $5^3 = 125$ in logarithmic form.

(b) Write the equation $12^2 = 144$ in logarithmic form.

(c) Write the equation $4^{-2} = \frac{1}{16}$ in logarithmic form.

(d) Write the equation $p^q = r$ in logarithmic form, where $p > 0$ and $p \neq 1$.

(e) Write the equation $e^y = 7$ in logarithmic form.

(a) Solve the equation $4 \log_2(3x) = 12$.

(b) Solve the equation $\ln(5-x) = 1$.

(c) Solve the equation $2 \log(\frac{x}{3}) = -4$.

(d) Solve the equation $\log_5(2x+1) + 1 = 3$.

Consider the functions $f(x) = e^{2x}$ and $g(x) = x+c$, where $c$ is a real constant.

(a) Write down an expression for $(f \circ g)(x)$.

(b) Given that $(f \circ g)(\ln 3) = 36$, find the value of $c$.

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 functions $f$ and $g$ are defined by

$f(x) = 2\ln x$, where $x > 0$

$g(x) = \ln(k(x-2) )$, where $x > 2, k \in \mathbb{R}^+$.

The graphs of $y = f(x)$ and $y = g(x)$ intersect at two distinct points.

(a) State the equation of the vertical asymptote to the graph of $y = g(x)$.

(b) (i) Show that, at the points of intersection, $x^2 - kx + 2k = 0$.

(b) (ii) Hence show that $k^2 - 8k > 0$.

(b) (iii) Find the range of possible values of $k$.

The graphs intersect at $x=p$ and $x=q$, where $p<q$.

(c) In the case where $k=10$, find the value of $q-p$. Express your answer in the form $a\sqrt{b}$, where $a, b \in \mathbb{Z}^+$.

Let $f(x) = \ln(1+ax)$, where $ax > -1, a \neq 0$.

The $n^{\text{th}}$ derivative of $f(x)$ is denoted by $f^{(n)}(x)$, for $n \in \mathbb{Z}^+$.

Prove by induction that $f^{(n)}(x) = (-1)^{n-1} (n-1)! a^n (1+ax)^{-n}$, for $n \in \mathbb{Z}^+$.

Hence or otherwise, find the Maclaurin series for $f(x) = \ln(1+ax)$ up to and including the $x^3$ term.

Hence, find the series expansion of $\ln\left(\frac{1+2x}{1-3x}\right)$ up to and including the $x^3$ term.

State the restriction which must be placed on $x$ for the approximation in part (c) to be valid.

Use a suitable value of $x$ to determine an approximate value for $\ln(2)$.

Give your answer as a rational number.

It is given that $\log_2(y) = -\frac{3}{2}$, where $y > 0$.

(a) Find the value of $\log_2(8y^2)$.

(b) Find the value of $\log_8(y)$.

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.

Consider the arithmetic sequence $c, d, e, ...$, where $c, d, e \neq 0$.

Show that $c + e = 2d$.

Consider the geometric sequence $c, f, g, ...$, where $c, f, g \neq 0$.

Show that $f^2 = cg$.

The first term of both sequences is $c$. It is given that $e = g = 4$ and that $f$ is a real number.

Show that $d > 2$.

Consider the case where $c = 16$, $f > 0$ and $e = g = 4$.

(i) Write down the first four terms of the arithmetic sequence.

(ii) Write down the first four terms of the geometric sequence.

A new sequence $v_n$ is formed by combining the terms of the arithmetic sequence, $A_n$, and the geometric sequence, $G_n$, from part (d). The terms of $v_n$ are given by $v_n = A_n - \log_2(G_n)$.

(i) Show that $v_n$ is an arithmetic sequence and find its common difference.

(ii) Hence, find the value of $\sum_{n=1}^{12} v_n$.

The growth of a bacterial colony, $B$, in a petri dish can be modelled by the logistic differential equation

$\frac{\text{d}B}{\text{d}t} = k B \left(1 - \frac{B}{N}\right)$

where $t$ is the time measured in hours and $k, N$ are positive constants.

The constant $N$ represents the maximum number of bacteria the petri dish can sustain indefinitely due to limited nutrients.

In the context of this bacterial growth model, interpret the meaning of $\frac{\text{d}B}{\text{d}t}$.

Show that $\frac{\text{d}^2B}{\text{d}t^2} = k^2B\left(1-\frac{B}{N}\right)\left(1-\frac{2B}{N}\right)$.

Hence show that the bacterial colony will grow at its maximum rate when $B = \frac{N}{2}$. Justify your answer.

Hence determine the maximum value of $\frac{\text{d}B}{\text{d}t}$ in terms of $k$ and $N$.

Let $B_0$ be the initial number of bacteria.

By solving the logistic differential equation, show that its solution can be expressed in the form

$kt = \ln\left(\frac{B(N-B_0)}{B_0(N-B)}\right)$.

After 5 hours, the number of bacteria is $2B_0$. It is known that $N = 3B_0$.

Find the value of $k$ for this bacterial growth model.

Differentiate each of the following expressions with respect to $x$:

$10e^x - 4\ln x + 2$

$e^{-5x}$

$x^2 e^x$

$\ln(4x^3 - 2x)$

$e^{\sin(x)}$

$\frac{\ln x}{x^3}$

$\ln(\frac{1}{x^2+1})$

$e^{x\cos(x)}$