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Exponential & Logarithms (Edexcel AS Maths: Pure): Exam Questions

Exam code: 8MA0

2 hours40 questions
Easy
Medium
Hard
1
Sme Calculator3 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?

23 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?

33 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
Sme Calculator3 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
Sme Calculator3 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
Sme Calculator1 mark

Solve the equation

2 to the power of x equals 16

How did you do?

7a
Sme Calculator2 marks

Solve the equation

x squared minus 12 x plus 27 equals 0

How did you do?

7b
Sme Calculator3 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
Sme Calculator2 marks

Solve the equation

2 log subscript 3 space 9 equals 5 x minus 6

How did you do?

92 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
Sme Calculator2 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
Sme Calculator2 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
Sme Calculator3 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
Sme Calculator4 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?

141 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?

151 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?

1a2 marks

Sketch the curve with equation

y equals 4 to the power of x

stating any points of intersection with the coordinate axes.

How did you do?

1b
Sme Calculator2 marks

Solve

4 to the power of x equals 100

giving your answer to 2 decimal places.

How did you do?

23 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?

3a3 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?

3b
Sme Calculator1 mark

Find the exact value of y spacewhen x equals 3.

How did you do?

4
Sme Calculator2 marks

Solve the equation

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

How did you do?

53 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?

6a3 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?

6b1 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?

7a1 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?

7b1 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?

7c
Sme Calculator3 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?

8a2 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?

8b2 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?

9a3 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?

9b1 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?

10
Sme Calculator3 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?

11a3 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?

11b1 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?

12a3 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?

12b1 mark

Find the value of x when y equals 0.064.

How did you do?

13a
Sme Calculator1 mark

Find the value of

log 1000 plus log 10000

How did you do?

13b
Sme Calculator1 mark

Solve

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

How did you do?

13c
Sme Calculator2 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?

14a3 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?

14b1 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?

15a
Sme Calculator2 marks

Solve

2 log 1000 equals x log subscript 16 4

How did you do?

15b
Sme Calculator2 marks

Solve

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

How did you do?

16a3 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?

16b1 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?

17a3 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?

17b1 mark

Find the value of x when y equals 625.

How did you do?

18a3 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?

18b
Sme Calculator2 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?

18c1 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
Sme Calculator3 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?

23 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?

3a2 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?

3b2 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
Sme Calculator3 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?

54 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?

63 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?

7a1 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
Sme Calculator2 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?