Laws of Logarithms (Cambridge International AS Mathematics): Exam Questions

Evaluate

(i) $\text{log}_{3} 27$

(ii) $\text{log}_{5} 625$

(iii) $\text{log}_{2} \frac{1}{4}$

(iv) $\text{log}_{a} a$

Write the following in the form $a + b \text{ln} 2$, where $a$ and $b$ are integers to be found.

(i) $3^{2} + \text{ln} 4$

(ii) $\text{ln} e^{7} + \text{ln} 8$

(iii) $\text{log} 1000 + 3 \text{ln} 16$

(iv) 5\left(3^{2} + \text{ln} 64\right)

Solve the following equations, giving your answer in exact form.

(i) $e^{2x} = 5$

(ii) $3 e^{\frac{1}{3}x} = 27$

Show that

$3 \text{log}_{a} 4 + 2 \text{log}_{a} 256 = 22 \text{log}_{a} 2$

Solve the equation

$\text{log}_{x} 16 = 2$

A square has side length $3 \text{ln} 4$.

Show that the perimeter of the square is $24 \text{ln} 2$.

Write the following in the form $a \text{ln} b$, where $a$ and $b$ are integers to be found.

$4 \text{ln} 9 + 2 \text{ln} 81 - 3 \text{ln} 27$

Solve the equation

$7^{2x-1} = 343$

Write down the value of

(i) $\text{log}_{3} 3$

(ii) $\text{ln} e^{6}$

(iii) $\text{log}_{a} 1$

(iv) $\text{log} 1000$

Show that

4\text{log}\left(\frac{27}{16}\right) = 12\text{log} 3 - 16\text{log} 2

Show that

$2 \text{ln} x^{3} - 3 \text{ln} x^{2} = 0$

Evaluate

$\text{log}_{2} 4 + \text{log}_{3} 27 - \text{log}_{4} 4$

Evaluate

$3 \text{ln} 2 + \frac{1}{2} \text{ln} 81 - 2 \text{ln} 3$

giving your answer in the form $\text{ln} q$, where $q$ is an integer to be found.

Solve the following equations, giving your answers in exact form.

$e^{x} = 5$

$3 e^{2x} = 9$

$e^{2x-1} = 4$

By writing $1 = \text{log}_{a} a$, show that

$1 + 2 \text{log}_{a} b + 3 \text{log}_{a} c = \text{log}_{a} a b^{2} c^{3}$

Write the following as a single logarithm

$2 \text{log}_{a} 6 + 3 \text{log}_{a} 2 - \text{log}_{a} 4$

Write the following in the form $a \text{ln} b$, where a and b are integers to be found.

$2 \text{ln} 3^{4} + \text{ln} 3^{3} - \text{ln} 9$

The diagram shows a triangle. The length of each of its three sides is measured in centimetres.

Work out the perimeter of the triangle, giving your answer in the form $2 \text{ln} b$, where $b$ is an integer to be found.

Solve the equation

$5^{2x} - 25 = 0$

Solve the equation

$3^{2x-1} = 4^{3} + 4^{2} + 1$

Solve the equation

\text{log}_{x}\left(5x - 6\right) = 2

Evaluate

$\text{log}_{2} 8^{2} + 3 \text{log}_{2} 16 - 2 \text{log}_{2} 2^{5}$

Evaluate

$3 \text{ln} 2 + 2 \text{ln} 5 - \frac{1}{2} \text{ln} 10000$

giving your answer in the form $\text{ln} p$.

By writing 5 as $5 \text{ln} e$, show that

$5 \text{ln} 2 + 5$

can be written as $5 \text{ln} 2 e$.

Evaluate

$4 \text{log}_{3} 729 + 3 \text{log}_{2} 64^{2} - 3 \text{log} 100 + \text{ln} e^{6}$

Evaluate

$\frac{1}{2} \text{ln} 196 + \frac{1}{3} \text{ln} 125 + \frac{1}{4} \text{ln} 81 + \frac{1}{5} \text{ln} 32$

giving your answer in the form $\text{ln} q$.

Solve the equation

$4^{3x+2} = 16^{x+6}$

Solve the equation

$4^{2x+3} - 8 = 92$

giving your answer correct to 3 significant figures.

Solve the following equations, giving your answers in exact form.

$4 e^{3x-2} = 12$

$3 e^{2x} + 8 = 14 e^{x}$

Simplify

$2 \text{ln} 3^{4} + \text{ln} 3^{3} - \text{ln} 9$

giving your answer in the form $a \text{ln} b$, where $a$ and $b$ are integers to be found.

Write

2\text{log}_{a} x + 3\text{log}_{a}\left(x + 1\right) - \text{log}_{a} 4\left(x + 2\right)

as a single logarithm.

Solve the equation

$5^{2x} - 8 \times 5^{x} + 12 = 0$

giving your answers in the form $\text{log}_{a} b$.

Solve the equation

$6 \times 3^{x-1} = 6^{2x}$

giving your answer in the form $\frac{\text{ln}a}{\text{ln}b}$, where $a$ and $b$ are integers to be found.

A ship sets sail from a harbour.

After some time, the ship's position is \left(4\text{ln} 3\right) km east of the harbour and \left(3\text{ln} 3\right) km north of the harbour.

Find the direct distance between the ship and the harbour at this time, giving your answer in the form \left(p\text{ln} 3\right) km, where $p$ is an integer to be found.

Solve the equation

\text{log}_{3}\left(x + 4\right) = 4 + 2\text{log}_{3} x

giving your answer correct to 3 significant figures.

Show that $4 - \text{ln} 16$ can be written in the form 4\text{ln}\left(\dfrac{e}{2}\right).

How many real solutions does the equation have? Justify your answer.

3\text{log}_{x}\left(x + 1\right) = \text{ln} e^{3}

Solve the equation

$2 \times 5^{2x+1} + 21 = 41 \times 5^{x}$

giving your answers in the form $\text{log}_{a} b$, where $a$ and $b$ are rational numbers to be found.

Show that

2\text{log}_{3} x + \text{log}_{3}\left(x^{2} - 1\right) - 2\text{log}_{3}\left(x + 1\right) \equiv \text{log}_{3}\frac{x^{2}\left(x - 1\right)}{\left(x + 1\right)}

Write the following as a single logarithm

2\text{log}_{p}\left(x + 1\right) + 3\text{log}_{p}\left(x - 1\right) - \text{log}_{p}\left(x^{2} - 1\right)

A triangle is drawn inside a circle such that one side of the triangle is the diameter and all three vertices of the triangle lie on the circumference.

The radius of the circle is \left(3\text{ln} 2\right) cm.

The two smallest angles in the triangle are $α$ and $β$ respectively, where $β = 2 α$.

Find all three sides of the triangle, giving your answers in the form $a \text{ln} 2$.

Without using a calculator, show that

$\text{log}_{4} 8 = \text{log}_{9} 27$