A LevelMathsOCRMaths APureExam QuestionsExponentials & LogarithmsExponential & Logarithms

Exponential & Logarithms (OCR A Level Maths A: Pure): Exam Questions

Exam code: H240

2 hours39 questions
Easy
Medium
Hard
1
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3 marks

Find the value of

(i) 3 cubed

(ii) 4 to the power of negative 2 end exponent

(iii) 9 to the power of 0.5 end exponent

How did you do?

2
3 marks

Sketch the curve with equation

y equals a to the power of x space space space space space space space space a greater than 1

showing clearly any points of intersection with the coordinate axes.

State whether the graph shows exponential growth or exponential decay.

How did you do?

3
3 marks

State whether the following models represent exponential growth or exponential decay:

(i) y equals 3 to the power of negative 2 x end exponent

(ii) y equals 20 open parentheses 2 close parentheses to the power of x

(iii) y equals 30 a to the power of negative x end exponent where a greater than 1

How did you do?

4
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3 marks

Write down the value(s) of a in the following:

(i) space 3 to the power of a equals 27

(ii) space a to the power of 1 third end exponent equals 5

(iii) space 4 a squared equals 64

How did you do?

5
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3 marks

Write down the value of a in the following:

(i) space log subscript 3 a equals 4

(ii) space log subscript a space end subscript 216 equals 3

(iii) space log subscript 2 space end subscript 128 equals a

How did you do?

6
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1 mark

Solve the equation

2 to the power of x equals 16

How did you do?

7a
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2 marks

Solve the equation

x squared minus 12 x plus 27 equals 0

How did you do?

7b
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3 marks

Hence solve the equation

open parentheses 3 to the power of x close parentheses squared minus 12 open parentheses 3 to the power of x close parentheses plus 27 equals 0

How did you do?

8
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2 marks

Solve the equation

2 log subscript 3 space 9 equals 5 x minus 6

How did you do?

9
2 marks

Sketch the curve with equation

y equals straight e to the power of x

showing clearly any points of intersection with the coordinate axes.

On your sketch, state the equation of the asymptote.

How did you do?

10
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2 marks

A curve has the equation y equals straight e to the power of 2 x end exponent

(i) Write down an expression for fraction numerator straight d y over denominator straight d x end fraction

(ii) Find the gradient of the point on the curve where x equals 0

How did you do?

11
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2 marks

Find the values of

(i) space 5 space log subscript 3 space 7

(ii) space 2 space log subscript 2 space 3 plus 3 space log subscript 3 space 2

giving your answers to 4 significant figures.

How did you do?

12
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3 marks

Solve the equation

straight e to the power of 2 x end exponent minus 16 equals 0

giving your answer in the form

a ln a 

where a is an integer.

How did you do?

13
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4 marks

Write down the value of a in the following:

(i) space log subscript a 8 equals 3

(ii) space log a equals 2

(iii) space ln open parentheses straight e cubed close parentheses equals a

(iv) space log subscript 5 a equals 1

How did you do?

14
1 mark

The function straight f open parentheses x close parentheses is defined by

straight f open parentheses x close parentheses equals 3 straight e to the power of 2 x end exponent space space space space space space space space x element of straight real numbers

Find an expression for straight f open parentheses 2 x close parentheses.

How did you do?

15
1 mark

The function straight f open parentheses x close parentheses is defined by

straight f open parentheses x close parentheses equals 5 straight e to the power of 3 x end exponent space space space space space space space space x element of straight real numbers

Find an expression for straight f open parentheses 4 x close parentheses.

How did you do?

1
3 marks

On the same set of coordinate axes, sketch the curves

y equals 2 to the power of x

and

y equals 3 to the power of x

Label clearly

  • any points where the curves meet the coordinate axes

  • the equation(s) of any asymptotes

How did you do?

2a
3 marks

Sketch the curve

y equals 1 over 2 to the power of x

State whether the graph shows exponential growth or exponential decay.

How did you do?

2b
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1 mark

Find the exact value of y spacewhen x equals 3.

How did you do?

3
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2 marks

Solve the equation

3 x plus 3 log subscript 2 4 equals 5 log subscript 6 216

How did you do?

4
3 marks

Use algebra to solve

2 to the power of 2 x end exponent minus 24 open parentheses 2 to the power of x close parentheses plus 128 equals 0

How did you do?

5a
3 marks

On the same set of coordinate axes, sketch the curves

y equals straight e to the power of x

and

y equals straight e to the power of negative x end exponent

Label clearly

  • any points where the curves meet the coordinate axes

  • the equation(s) of any asymptotes

How did you do?

5b
1 mark

Write down the gradient of the curve

y equals straight e to the power of x

at the point open parentheses 0 comma 1 close parentheses.

How did you do?

6a
1 mark

Given that

y equals straight e to the power of 4 x end exponent

write down an expression for fraction numerator straight d y over denominator straight d x end fraction

How did you do?

6b
1 mark

Given that

y equals 2 straight e to the power of 2 x end exponent

write down an expression for fraction numerator straight d y over denominator straight d x end fraction

How did you do?

6c
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3 marks

A curve has equation

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

Find the gradient of the point on the curve where x equals 3.

Give your answer in the form

p straight e to the power of q

where p and q are integers to be found.

How did you do?

7a
2 marks

(i) Write down the gradient function of the curve y equals straight e to the power of negative 3 x end exponent

(ii) Find the gradient of the curve y equals straight e to the power of negative 3 x end exponent at the point where x equals 0

How did you do?

7b
2 marks

(i) write down the exact gradient of the point on the curve y equals straight e to the power of negative 3 x end exponent where x equals 2

(ii) Find the x-coordinate of the point on the curve y equals straight e to the power of negative 3 x end exponent which has a gradient of negative 3 straight e to the power of negative 12 end exponent

How did you do?

8a
3 marks

The function straight f open parentheses x close parentheses is defined by

straight f open parentheses x close parentheses equals 2 straight e to the power of 3 x end exponent space space space space space space space space x element of straight real numbers

(i) Write down the function straight f open parentheses negative x close parentheses.

(ii) On the same set of coordinate axes, sketch the curves of y equals straight f open parentheses x close parentheses and y equals straight f open parentheses negative x close parentheses.

Label any points where the graphs intersect the coordinate axes.

How did you do?

8b
1 mark

A different function is given by

straight g open parentheses x close parentheses equals negative 2 straight e to the power of 3 x end exponent

State the single graph transformation that maps the curve y equals straight f open parentheses x close parentheses to the curve y equals straight g open parentheses x close parentheses.

How did you do?

9
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3 marks

Solve

straight e to the power of 2 x end exponent minus 8 straight e to the power of x plus 15 equals 0

giving your answer(s) to 3 significant figures.

How did you do?

10a
3 marks

On the same set of coordinate axes, sketch the curves

y equals 4 to the power of x

and

y equals 5 to the power of x

Label clearly

  • any points where the curves meet the coordinate axes

  • the equation(s) of any asymptotes

How did you do?

10b
1 mark

Write down an equation of the curve that is a reflection of y equals 4 to the power of x in the y-axis.

How did you do?

11a
3 marks

(i) Sketch the curve with equation y equals 0.4 to the power of x

(ii) State whether the curve represents exponential growth or exponential decay.

How did you do?

11b
1 mark

Find the value of x when y equals 0.064.

How did you do?

12a
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1 mark

Find the value of

log 1000 plus log 10000

How did you do?

12b
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1 mark

Solve

6 to the power of log subscript 6 space a end exponent equals 36

How did you do?

12c
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2 marks

Find the value of

fraction numerator 2 log subscript 4 64 plus 3 to the power of log subscript 2 8 end exponent minus log subscript 5 5 over denominator log 100 end fraction

How did you do?

13a
3 marks

Sketch the curve with equation

y equals 12 straight e to the power of negative x end exponent space space space space space space space space space x greater or equal than 0

stating any points of intersection with the coordinate axes.

Write down the equations of any asymptotes.

How did you do?

13b
1 mark

Find the gradient of the point on the curve y equals 12 straight e to the power of negative x end exponent where x equals 0.

How did you do?

14a
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2 marks

Solve

2 log 1000 equals x log subscript 16 4

How did you do?

14b
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2 marks

Solve

3 log subscript 4 x equals log subscript 4 x plus 3 log subscript 5 2 5

How did you do?

15a
3 marks

On the same set of coordinate axes, sketch the curves

y equals 0.3 to the power of x

and

y equals 0.5 to the power of x

Label clearly

  • any points where the curves meet the coordinate axes

  • the equation(s) of any asymptotes

How did you do?

15b
1 mark

Write down an equation of the curve that is a reflection of y equals 0.5 to the power of x in the y-axis.

How did you do?

16a
3 marks

Sketch the curve

y equals 0.2 to the power of negative x end exponent

State

  • the equation of the asymptote

  • whether the graph shows exponential growth or exponential decay.

How did you do?

16b
1 mark

Find the value of x when y equals 625.

How did you do?

17a
3 marks

Sketch the curve with equation

y equals 4 straight e to the power of x space space space space space space space space space x greater or equal than 0

stating any points of intersection with the coordinate axes.

Write down the equations of any asymptotes.

How did you do?

17b
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2 marks

Find the gradient of the point on the curve y equals 4 straight e to the power of x where x equals 3.

Give your answer correct to 3 significant figures.

How did you do?

17c
1 mark

The growth of a population, P, at time t years is modelled by the equation

P equals 4 straight e to the power of t


Write down the initial population.

How did you do?

1
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3 marks

Solve

2 straight e to the power of 2 x end exponent equals straight e to the power of x plus 10

giving your answer(s) to 3 significant figures.

How did you do?

2
3 marks

Solve

2 open parentheses 2 to the power of 2 x end exponent close parentheses plus 4 equals 9 open parentheses 2 to the power of x close parentheses

How did you do?

3a
2 marks

Show that the equation

straight e to the power of x minus straight e to the power of negative x end exponent equals 0

has only one real solution, which you should find.

How did you do?

3b
2 marks

Show that the equation

straight e to the power of x plus straight e to the power of negative x end exponent equals 0

has no real solutions.

How did you do?

4
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3 marks

Without using a calculator, find the value of

fraction numerator 3 log subscript 6 216 minus ln open parentheses straight e to the power of 5 close parentheses plus 4 to the power of log subscript 5 625 end exponent over denominator log 10000 end fraction

Show the steps in your solution clearly.

How did you do?

5
4 marks

Solve

3 to the power of 2 open parentheses x plus 1 close parentheses end exponent plus 3 equals 28 open parentheses 3 to the power of x close parentheses

How did you do?

6
3 marks

Find two values of x for which

log open parentheses x squared close parentheses equals open parentheses log space x close parentheses squared

is true.

How did you do?

7a
1 mark

A curve has equation

y equals 1 over a straight e to the power of negative b x end exponent

where a and b are non-zero constants.

State a condition on b to ensure the curve represents exponential decay.

How did you do?

7b
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2 marks

At the point open parentheses 0 comma space a close parentheses on the curve, the gradient is 10.

Use this information to find an expression for y in terms of a and x only.

How did you do?