A LevelMathsCambridge (CIE)Pure 3Exam QuestionsLogarithms & ExponentialsLaws of Logarithms

Laws of Logarithms (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

Exam code: 9709

2 hours35 questions
Easy
Medium
Hard
Very Hard
1
4 marks

Evaluate

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

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

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

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

How did you do?

2
4 marks

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)

How did you do?

3
4 marks

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

(i) e^{2x} = 5

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

How did you do?

4
3 marks

Show that

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

How did you do?

5
2 marks

Solve the equation

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

How did you do?

6
2 marks

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

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

How did you do?

7
3 marks

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

How did you do?

8
2 marks

Solve the equation

7^{2x - 1} = 343

How did you do?

9
4 marks

Write down the value of

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

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

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

(iv) \text{log} 1000

How did you do?

10
3 marks

Show that

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

How did you do?

11
3 marks

Show that

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

How did you do?

1a
2 marks

Evaluate

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

How did you do?

1b
3 marks

Evaluate

3\text{ln} 2 + \tfrac{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.

How did you do?

2a
1 mark

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

e^{x} = 5

How did you do?

2b
3 marks

3e^{2x} = 9

How did you do?

2c
3 marks

e^{2x - 1} = 4

How did you do?

3
3 marks

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

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

How did you do?

4a
3 marks

Write the following as a single logarithm

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

How did you do?

4b
3 marks

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

How did you do?

5
4 marks

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

Triangle with its three sides labelled 4 ln 2, 3 ln 4 and 2 ln 3 centimetres, marked not to scale

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.

How did you do?

6a
2 marks

Solve the equation

5^{2x} - 25 = 0

How did you do?

6b
2 marks

Solve the equation

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

How did you do?

7
3 marks

Solve the equation

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

How did you do?

8a
2 marks

Evaluate

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

How did you do?

8b
3 marks

Evaluate

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

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

How did you do?

9
3 marks

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

5\text{ln} 2 + 5

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

How did you do?

10a
2 marks

Evaluate

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

How did you do?

10b
3 marks

Evaluate

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

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

How did you do?

1a
2 marks

Solve the equation

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

How did you do?

1b
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3 marks

Solve the equation

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

giving your answer correct to 3 significant figures.

How did you do?

2a
2 marks

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

4e^{3x - 2} = 12

How did you do?

2b
3 marks

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

How did you do?

3a
2 marks

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.

How did you do?

3b
2 marks

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.

How did you do?

4
3 marks

Solve the equation

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

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

How did you do?

5
5 marks

Solve the equation

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

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

How did you do?

6
4 marks

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.

How did you do?

7
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3 marks

Solve the equation

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

giving your answer correct to 3 significant figures.

How did you do?

8
3 marks

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

How did you do?

9
3 marks

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

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

How did you do?

1
4 marks

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.

How did you do?

2
3 marks

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)}

How did you do?

3
3 marks

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)

How did you do?

4
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5 marks

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 \alpha and \beta respectively, where \beta = 2\alpha.

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

How did you do?

5
3 marks

Without using a calculator, show that

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

How did you do?