A LevelMathsEdexcelPureExam QuestionsExponentials & LogarithmsLaws of Logarithms

Laws of Logarithms (Edexcel A Level Maths: Pure): Exam Questions

Exam code: 9MA0

2 hours42 questions
Easy
Medium
Hard
Very Hard
1
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4 marks

Find the value of

(i) log subscript 3 27

(ii) log subscript 5 625

(iii) log subscript 2 1 fourth

(iv) log subscript a space a where a greater than 0

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2
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4 marks

Write the following in the form a plus b space ln space 2, where a spaceand b are integers to be found:

(i) 3 squared plus ln space 4

(ii) ln open parentheses straight e to the power of 7 close parentheses plus ln 8

(iii) log 1000 plus 3 ln 16

(iv) 5 open parentheses 3 squared plus ln 64 close parentheses

How did you do?

3
3 marks

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

(i) straight e to the power of 2 x end exponent equals 5

(ii) 3 straight e to the power of 1 third x end exponent equals 27

How did you do?

4
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2 marks

Show that

3 log subscript a 4 plus 2 log subscript a 256

can be written as

22 log subscript a 2

where a greater than 0.

How did you do?

5
1 mark

Solve the equation

log subscript x 16 equals 2

How did you do?

6
2 marks

A square has a side length of 3 ln space 4

Find the perimeter of the square, giving your answer in the form

k ln space 2

where k is an integer to be found.

How did you do?

7
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2 marks

Express

4 ln 9 plus 2 ln 81 minus 3 ln 27

in the form

a ln 3

where a spaceis an integer to be found.

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8
2 marks

Solve the equation

7 to the power of 2 x minus 1 end exponent equals 343

How did you do?

9
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4 marks

Write down the value of

(i) log subscript 3 3

(ii) ln open parentheses straight e to the power of 6 close parentheses

(iii) log subscript a 1 where a greater than 0

(iv) log 1000

How did you do?

10
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3 marks

Show that

4 log open parentheses 27 over 16 close parentheses

can be expressed in the form

12 log 3 minus 16 log 2

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11a
1 mark

Express 42 as a product of its prime factors.

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11b
2 marks

Hence show that

ln 42 equals ln p plus ln q plus ln r

where p, q and r are distinct prime numbers.

How did you do?

12a
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2 marks

Find the value of

log subscript 2 4 plus log subscript 3 27 minus log subscript 4 4

How did you do?

12b
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2 marks

Express

3 ln 2 plus 1 half ln 81 minus 2 ln 3

in the form ln space q, where q is an integer to be found.

How did you do?

13a
1 mark

Solve the equation

straight e to the power of x equals 5

giving your answer in an exact form.

How did you do?

13b
2 marks

Solve

3 straight e to the power of 2 x end exponent equals 9

giving your answer in an exact form.

How did you do?

13c
2 marks

Solve

straight e to the power of 2 x minus 1 end exponent equals 4

giving your answer in an exact form.

How did you do?

14a
3 marks

Express

2 log subscript a 6 plus 3 log subscript a 2 minus log subscript a 4

in the form

log subscript a k

where a greater than 0 and k is an integer to be found.

How did you do?

14b
3 marks

Express

2 ln open parentheses 3 to the power of 4 close parentheses plus ln open parentheses 3 cubed close parentheses minus ln 9

in the form

a ln 3

where a is an integer to be found.

How did you do?

15a
1 mark

Solve the equation

5 to the power of 2 x end exponent minus 25 equals 0

How did you do?

15b
2 marks

Solve the equation

3 to the power of 2 x minus 1 end exponent equals 4 cubed plus 4 squared plus 1

How did you do?

16a
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2 marks

Express

log subscript 2 open parentheses 8 squared close parentheses plus 3 log subscript 2 16 minus 2 log subscript 2 open parentheses 2 to the power of 5 close parentheses

as an integer.

How did you do?

16b
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3 marks

Express

3 ln 2 plus 2 ln 5 minus 1 half ln 10000

in the form ln space p where p is an integer.

How did you do?

17
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2 marks

Without using a calculator, show that

log subscript 4 8 equals log subscript 9 27

How did you do?

1a
3 marks

Given that

2 log open parentheses 4 minus x close parentheses equals log open parentheses x plus 8 close parentheses

show that

x squared minus 9 x plus 8 equals 0

How did you do?

1b
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2 marks

(i) Write down the roots of the equation

x squared minus 9 x plus 8 equals 0

(ii) State which of the roots in (b)(i) is not a solution of

2 log left parenthesis 4 minus x right parenthesis equals log left parenthesis x plus 8 right parenthesis

giving a reason for your answer.

How did you do?

2
3 marks

Using the laws of logarithms, solve the equation

log subscript 3 open parentheses 12 y plus 5 close parentheses minus log subscript 3 open parentheses 1 minus 3 y close parentheses equals 2

How did you do?

3a
3 marks

Given that a greater than b greater than 0 and that a and b satisfy the equation

log space a minus log space b equals log open parentheses a minus b close parentheses

show that

a equals fraction numerator b squared over denominator b minus 1 end fraction

How did you do?

3b
2 marks

Write down the full restriction on the value of b, explaining the reason for this restriction.

How did you do?

4
2 marks

Express

1 plus 2 log subscript a b plus 3 log subscript a c

in the form

log subscript a space y

where a comma space b comma space c greater than 0 and y is an expression in terms of a, b and c that you should find.

How did you do?

5
2 marks

The diagram below shows the length of three sides of a triangle.

q5-6-1-laws-of-logarithms-edexcel-a-level-pure-maths-medium

Find the perimeter of the triangle, giving your answer in the form

2 ln space b

where b is an integer to be found.

How did you do?

6
2 marks

Simplify fully the expression

2 ln open parentheses x cubed close parentheses minus 3 ln open parentheses x squared close parentheses

How did you do?

7
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3 marks

Solve the equation

4 to the power of 2 x plus 3 end exponent minus 8 equals 92

giving your answer to 3 significant figures.

How did you do?

8
4 marks

(i) On the same set of coordinate axes, sketch the curves y equals straight e to the power of x and y equals ln space x.

Label clearly the coordinates of any points where the curves meet the coordinate axes.

(ii) State the equation of any asymptotes and the equation of the line of reflection between the two curves.

How did you do?

9
2 marks

A ship sets sail from a harbour.

After some time, the ship’s position is open parentheses 4 ln 3 close parentheses km due east of the harbour and open parentheses 3 ln 3 close parentheses km due north of the harbour.

Find the distance between the ship and the harbour at this time, giving your answer in the form open parentheses p ln 3 close parentheses km.

How did you do?

10
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2 marks

Express

5 ln 2 plus 5

in the form 5 ln k where k is an irrational number to be found.

How did you do?

11
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3 marks

Solve

8 straight e to the power of open parentheses 3 x squared minus 1 close parentheses end exponent equals 12

giving your answer(s) to 3 significant figures.

How did you do?

1
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3 marks

By taking logarithms of both sides, solve the equation

4 to the power of 3 p minus 1 end exponent equals 5 to the power of 210

giving the value of p to one decimal place.

How did you do?

2a
2 marks

Given that p equals log subscript 3 x, where x greater than 0, find in simplest form in terms of p,

(i) log subscript 3 open parentheses x over 9 close parentheses

(ii) log subscript 3 open parentheses square root of x close parentheses

How did you do?

2b
4 marks

Hence, or otherwise, solve

2 log subscript 3 open parentheses x over 9 close parentheses plus 3 log subscript 3 open parentheses square root of x close parentheses equals negative 11

giving your answer as a simplified fraction.

Solutions relying on calculator technology are not acceptable.

How did you do?

3
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5 marks

Using the laws of logarithms, solve the equation

2 log subscript 5 open parentheses 3 x minus 2 close parentheses minus log subscript 5 x equals 2

How did you do?

4
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3 marks

Solve the equation

3 straight e to the power of 2 x end exponent plus 8 equals 14 straight e to the power of x

giving your answer(s) in the form ln space a where a is a constant to be found.

How did you do?

5
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3 marks

Express

2 log x plus 3 log open parentheses x plus 1 close parentheses minus log open parentheses 4 open parentheses x plus 2 close parentheses close parentheses

in the form

log straight f open parentheses x close parentheses

where straight f open parentheses x close parentheses is a function that you should state.

How did you do?

6
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3 marks

Solve the equation

5 to the power of 2 x end exponent minus 8 cross times 5 to the power of x plus 12 equals 0

giving your answer(s) in the form log subscript 5 space end subscript k where k is a constant to be found.

How did you do?

7
5 marks

Solve the equation

6 cross times 3 to the power of x minus 1 end exponent equals 6 to the power of 2 x end exponent

giving your answer in the form

fraction numerator ln a over denominator ln b end fraction

where a and b are positive integers to be found.

How did you do?

8
3 marks

Solve the equation

log subscript x open parentheses 5 x minus 6 close parentheses equals 2

How did you do?

9
4 marks

Solve the equation

2 cross times 5 to the power of 2 x plus 1 end exponent plus 21 equals 41 cross times 5 to the power of x

giving your answer(s) in the form log subscript 5 a, where a is a constant to be found.

How did you do?

10
3 marks

Show that

2 log subscript 3 x plus log subscript 3 open parentheses x squared minus 1 close parentheses minus 2 log subscript 3 open parentheses x plus 1 close parentheses

can be expressed in the form

log subscript 3 open parentheses fraction numerator x squared open parentheses x minus 1 close parentheses over denominator x plus 1 end fraction close parentheses

where x greater than 1.

How did you do?

1
3 marks

Given that I equals integral subscript 3 superscript 9 log subscript 3 open parentheses 2 x close parentheses space straight d x, write each of the following in terms of I

(i) space integral subscript 3 superscript 9 log subscript 3 open parentheses 2 x close parentheses to the power of 10 space straight d x

(ii) space integral subscript 3 superscript 9 log subscript 3 open parentheses 18 x close parentheses space straight d x

How did you do?

2a
3 marks

Given that

log subscript 2 open parentheses x plus 3 close parentheses plus log subscript 2 open parentheses x plus 10 close parentheses equals 2 plus 2 log subscript 2 x

show that

3 x squared minus 13 x minus 30 equals 0

How did you do?

2b
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2 marks

(i) Write down the roots of the equation

3 x squared minus 13 x minus 30 equals 0

(ii) Hence state which of the roots in part (b)(i) is not a solution of

log subscript 2 open parentheses x plus 3 close parentheses plus log subscript 2 open parentheses x plus 10 close parentheses equals 2 plus 2 log subscript 2 x

giving a reason for your answer.

How did you do?

3
4 marks

The curve with equation y equals 3 cross times 2 to the power of x meets the curve with equation y equals 15 minus 2 to the power of x plus 1 end exponent at the point P.

Find, using algebra, the exact x coordinate of P.

How did you do?

4a
1 mark

a equals log subscript 2 x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space b equals log subscript 2 open parentheses x plus 8 close parentheses

Express in terms of a and/or b

log subscript 2 square root of x

How did you do?

4b
2 marks

log subscript 2 open parentheses x squared plus 8 x close parentheses

How did you do?

4c
3 marks

log subscript 2 open parentheses 8 plus 64 over x close parentheses

Give your answer in simplest form.

How did you do?