Browse Exponentials & logarithms

134 questions at your level

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

Exponential functions and e

11 questions

LessonNot started

e^{2x} = 5

, giving your answer in exact form.
(b) Explain why the equation

e^{2x} = -1

has no real solutions.

●●●●●Level 33 marksStart

Solve the equation

\mathrm{e}^{2x} - 7\mathrm{e}^{x} + 10 = 0

giving your answers exactly. (4 marks)

●●●●●Level 34 marksStart

A company launches a new online game. An analyst proposes two possible models for the number of active players,

P

thousand,

t

months after the launch. The diagrams show these models as they apply to the first 12 months.

Model AModel B(a) Without calculation, describe briefly how the rate of growth of

P

varies during the first 12 months, according to each of these two models.

(1 mark)

The equation of the curve for model A is

P = 20 + 30te^{-0.25t}

. The equation of the curve for model B is

P = 80 - 60e^{-0.3t}

.
(b) Describe the behaviour of

P

that is predicted for

t > 12

(i) using model A,

(ii) using model B.

(2 marks)

The company will stop running the game if, at any time after the first year, the number of active players falls below 25 thousand.
(c) State, with a reason, which model predicts that the company will eventually stop running the game.

(1 mark)

●●●●●Level 34 marksStart

Logarithms

13 questions

LessonNot started

Find the value of

\log_8 32

●●●●●Level 32 marksStart

The equation

x^3 + 3\ln x = 5

has exactly one real root.
(a) Show by calculation that the root lies between

1.54

and

1.55

.

(2 marks)

In order to find the root, the iterative formula

x_{n+1} = \left(p + q\ln x_n\right)^r,

with initial value

x_0 = 1.5

, is to be used.
(b) Determine the values of the constants

p

q

and

r

.

(2 marks)
(c) Hence find the root correct to 4 significant figures. Show the result of each step of the iteration process.

(2 marks)

●●●●●Level 36 marksStart

Prove by contradiction that

\log_2 3

is irrational. (4 marks)

●●●●●Level 34 marksStart

Given that

\log_3 x^2 - \log_3 y^5 = 4

show that

x = Ay^p

where

A

is an integer and

p

is a rational number.

(4 marks)

●●●●●Level 34 marksStart

The laws of logarithms

45 questions

LessonNot started

Given that

\log_2 5 = 2.32

to 3 significant figures, find

\log_2 40

to 3 significant figures, without a calculator's log button.

●●●●●Level 32 marksStart

In this question you must show detailed reasoning.

Solve the equation

\log_2(x + 5) + \log_2(x - 2) = 3

.

(4 marks)

●●●●●Level 34 marksStart

A botanist measures the area,

A

cm², covered by a patch of lichen on a stone wall on 1st March each year. She thinks that the area can be modelled by the equation

A = ab^t

, where

a

and

b

are both positive constants and

t

is the number of years since 1st March 2019.

To test this, she uses the areas measured on 1st March in each of the years 2019 to 2025 and plots

\log_{10} A

against

t

.
(a) Explain why, if the botanist's model is correct, the results should lie on a straight line of best fit on her graph.

(3 marks)

The results lie on a straight line of best fit which has a gradient of

0.0842

and an intercept on the vertical axis of

1.37

.
(b) Use these values to obtain estimates for

a

and

b

, correct to 2 significant figures.

(2 marks)

●●●●●Level 35 marksStart

A continuous curve has equation

y = \log_3(2x + 5)

The table below shows corresponding values of

x

and

y

for this curve, with the values of

y

given to 4 decimal places as appropriate.

\begin{array}{c|cccccc} x & 0 & 0.4 & 0.8 & 1.2 & 1.6 & 2 \\ \hline y & 1.4650 & 1.6001 & 1.7177 & 1.8218 & 1.9153 & 2 \end{array}

Using the trapezium rule with all the values of

y

in the given table,
(a) find an estimate for

\int_{0}^{2} \log_3(2x + 5)\,\mathrm{d}x

giving your answer to 3 decimal places.

(3 marks)
(b) Using your answer to part (a) and making your method clear, estimate

(i)

\displaystyle\int_{0}^{2} \log_3\sqrt{2x + 5}\,\mathrm{d}x

(ii)

\displaystyle\int_{0}^{2} \log_3(18x + 45)\,\mathrm{d}x

(2 marks)

●●●●●Level 35 marksStart

Solve the equation

3^{2x+1} = 5^{x+2}

,

giving your answer in the form

x = \dfrac{\ln a}{\ln b}

, where

a

and

b

are rationals, and then to 3 significant figures. (5 marks)

●●●●●Level 35 marksStart

Solve the equation

\log_3(2x + 1) + \log_3(x - 1) = 2

You must show clearly why any value you obtain must be rejected. (5 marks)

●●●●●Level 35 marksStart

Solve the equation

2\log_2 x - \log_2 (x + 3) = 1

giving your answer in exact form. Fully justify your answer. (5 marks)

●●●●●Level 35 marksStart

Solve the equation

\ln(2x+3) = 2\ln x

. (4 marks)

●●●●●Level 34 marksStart

Solve the equation

2\log_3 x - \log_3 (x + 6) = 1

,

explaining why one root of your quadratic must be rejected. (5 marks)

●●●●●Level 35 marksStart

Given that

\log_3 x^2 - \log_3 y^5 = 4

show that

x = Ay^p

where

A

is an integer and

p

is a rational number.

(4 marks)

●●●●●Level 34 marksStart

Given that

\log_5(x + 6) + \log_5(x - 6) = 1 + \log_5 x

(a) show that

x^2 - 5x - 36 = 0

(3 marks)
(b) (i) Write down the roots of the equation

x^2 - 5x - 36 = 0

(ii) Hence state, giving a reason for your answer, which of these roots is not a solution of

\log_5(x + 6) + \log_5(x - 6) = 1 + \log_5 x

(2 marks)

●●●●●Level 35 marksStart

Given that

\log_8(x + 6) + \log_8(x - 1) = 1

(a) show that

x^2 + 5x - 14 = 0

(3 marks)
(b) (i) Write down the roots of the equation

x^2 + 5x - 14 = 0

(ii) Hence state, giving a reason for your answer, which of these roots is not a solution of

\log_8(x + 6) + \log_8(x - 1) = 1

(2 marks)

●●●●●Level 35 marksStart

Solving exponential and logarithmic equations

57 questions

LessonNot started

Solve

e^{2x} - 7e^{x} + 12 = 0

, giving your answers in exact form.

●●●●●Level 33 marksStart

Solve

2^{2x} - 9(2^{x}) + 8 = 0

●●●●●Level 33 marksStart

Solve

7^{x} = 3^{x + 1}

, giving your answer to 3 significant figures.

●●●●●Level 33 marksStart

In this question you must show detailed reasoning.

Solve the equation

\log_2(x + 5) + \log_2(x - 2) = 3

.

(4 marks)

●●●●●Level 34 marksStart

Solve the equation

3^{2x+1} = 5^{x+2}

,

giving your answer in the form

x = \dfrac{\ln a}{\ln b}

, where

a

and

b

are rationals, and then to 3 significant figures. (5 marks)

●●●●●Level 35 marksStart

Solve the equation

\log_3(2x + 1) + \log_3(x - 1) = 2

You must show clearly why any value you obtain must be rejected. (5 marks)

●●●●●Level 35 marksStart

Solve the equation

2\log_2 x - \log_2 (x + 3) = 1

giving your answer in exact form. Fully justify your answer. (5 marks)

●●●●●Level 35 marksStart

Solve the equation

\ln(2x+3) = 2\ln x

. (4 marks)

●●●●●Level 34 marksStart

Solve the equation

2\log_3 x - \log_3 (x + 6) = 1

,

explaining why one root of your quadratic must be rejected. (5 marks)

●●●●●Level 35 marksStart

Solve the equation

3^{2x + 1} - 10\left(3^{x}\right) + 3 = 0

giving your answers in exact form. (4 marks)

●●●●●Level 34 marksStart

A geometric series has first term

5

and common ratio

1.2

.
(a) Find the

12

th term of the series, giving your answer to

3

significant figures. (2 marks)
(b) Find the least value of

n

for which the sum of the first

n

terms exceeds

1000

. (3 marks)

●●●●●Level 35 marksStart

The temperature

\theta

°C of a machine component,

t

minutes after the machine is switched on, is modelled by

\theta = 80 - 55\mathrm{e}^{-0.08t}

(a) State the temperature of the component at the moment the machine is switched on. (1 mark)
(b) Find the time taken for the component to reach

60

°C, giving your answer to

1

decimal place. (3 marks)
(c) Explain why this model predicts that the temperature never reaches

80

°C. (2 marks)

●●●●●Level 36 marksStart

Given that

\log_5(x + 6) + \log_5(x - 6) = 1 + \log_5 x

(a) show that

x^2 - 5x - 36 = 0

(3 marks)
(b) (i) Write down the roots of the equation

x^2 - 5x - 36 = 0

(ii) Hence state, giving a reason for your answer, which of these roots is not a solution of

\log_5(x + 6) + \log_5(x - 6) = 1 + \log_5 x

(2 marks)

●●●●●Level 35 marksStart

Given that

\log_8(x + 6) + \log_8(x - 1) = 1

(a) show that

x^2 + 5x - 14 = 0

(3 marks)
(b) (i) Write down the roots of the equation

x^2 + 5x - 14 = 0

(ii) Hence state, giving a reason for your answer, which of these roots is not a solution of

\log_8(x + 6) + \log_8(x - 1) = 1

(2 marks)

●●●●●Level 35 marksStart

Exponential growth and decay models

48 questions

LessonNot started

A bacterial culture grows according to

N = N_0\,e^{kt}

, where

t

is in hours. Initially there are

500

bacteria, and after

4

hours there are

1500

.
(a) Find

N_0

and the value of

k

to 3 significant figures.
(b) Find the number of bacteria after

6

hours, to the nearest hundred.

●●●●●Level 35 marksStart

A town's population is modelled by

P = A\,e^{kt}

, where

t

is the number of years after the year 2000. In 2002 the population was

1200

, and in 2005 it was

1800

. Find the values of

A

and

k

, giving each to 3 significant figures.

●●●●●Level 34 marksStart

A radioactive sample has mass

m = 80 e^{-0.025t}

grams, where

t

is the time in years.
(a) State the initial mass. (1 mark)
(b) Find the half-life of the sample, to 3 s.f. (3 marks)
(c) Find the mass remaining after

50

years, to 3 s.f. (2 marks)

●●●●●Level 36 marksStart

A botanist measures the area,

A

cm², covered by a patch of lichen on a stone wall on 1st March each year. She thinks that the area can be modelled by the equation

A = ab^t

, where

a

and

b

are both positive constants and

t

is the number of years since 1st March 2019.

To test this, she uses the areas measured on 1st March in each of the years 2019 to 2025 and plots

\log_{10} A

against

t

0.0842

and an intercept on the vertical axis of

1.37

.
(b) Use these values to obtain estimates for

a

and

b

, correct to 2 significant figures.

(2 marks)

●●●●●Level 35 marksStart

A company launches a new online game. An analyst proposes two possible models for the number of active players,

P

thousand,

t

months after the launch. The diagrams show these models as they apply to the first 12 months.

Model AModel B(a) Without calculation, describe briefly how the rate of growth of

P

varies during the first 12 months, according to each of these two models.

(1 mark)

The equation of the curve for model A is

P = 20 + 30te^{-0.25t}

. The equation of the curve for model B is

P = 80 - 60e^{-0.3t}

.
(b) Describe the behaviour of

P

that is predicted for

t > 12

●●●●●Level 34 marksStart

The number of mussels on a new sea wall is being monitored.

The number of mussels,

N

, on the sea wall,

t

months after monitoring began, is modelled by the equation

\log_{10} N = 0.25t + 3

Use the equation of the model to answer parts (a) to (c).
(a) Find the initial number of mussels on the sea wall.

(1 mark)
(b) Show that the equation of the model can be written in the form

N = ab^t

where

a

and

b

are constants to be found. Give the value of

b

to 2 decimal places.

(3 marks)
(c) Hence find the rate of increase of the number of mussels on the sea wall exactly

4

months after monitoring began.

(2 marks)

●●●●●Level 36 marksStart

The mass

m

mg of a radioactive isotope in a sample after

t

days is modelled by

m = 240e^{-kt}

,

where

k

is a positive constant. When

t = 8

, the mass has fallen to

180

mg.
(a) Show that

k = \frac{1}{8}\ln\frac{4}{3}

, and evaluate

k

to 3 significant figures. (3 marks)
(b) Find the mass predicted by the model after 30 days. (1 mark)
(c) Find, to 3 significant figures, the number of days it takes for the mass to halve. (2 marks)
(d) Find the rate at which the mass is decreasing when

t = 10

. (2 marks)

●●●●●Level 38 marksStart

The temperature

\theta

°C of a machine component,

t

minutes after the machine is switched on, is modelled by

\theta = 80 - 55\mathrm{e}^{-0.08t}

(a) State the temperature of the component at the moment the machine is switched on. (1 mark)
(b) Find the time taken for the component to reach

60

°C, giving your answer to

1

decimal place. (3 marks)
(c) Explain why this model predicts that the temperature never reaches

80

°C. (2 marks)

●●●●●Level 36 marksStart

Logarithms and non-linear data

20 questions

LessonNot started

The variables

x

and

y

are related by

y = ax^{n}

. When

x = 2

y = 10

, and when

x = 8

y = 80

. Find the values of

n

and

a

, giving

a

to 3 significant figures.

●●●●●Level 34 marksStart

A botanist measures the area,

A

cm², covered by a patch of lichen on a stone wall on 1st March each year. She thinks that the area can be modelled by the equation

A = ab^t

, where

a

and

b

are both positive constants and

t

is the number of years since 1st March 2019.

To test this, she uses the areas measured on 1st March in each of the years 2019 to 2025 and plots

\log_{10} A

against

t

0.0842

and an intercept on the vertical axis of

1.37

.
(b) Use these values to obtain estimates for

a

and

b

, correct to 2 significant figures.

(2 marks)

●●●●●Level 35 marksStart

The number of mussels on a new sea wall is being monitored.

The number of mussels,

N

, on the sea wall,

t

months after monitoring began, is modelled by the equation

\log_{10} N = 0.25t + 3

Use the equation of the model to answer parts (a) to (c).
(a) Find the initial number of mussels on the sea wall.

(1 mark)
(b) Show that the equation of the model can be written in the form

N = ab^t

where

a

and

b

are constants to be found. Give the value of

b

to 2 decimal places.

(3 marks)
(c) Hence find the rate of increase of the number of mussels on the sea wall exactly

4

months after monitoring began.

(2 marks)

●●●●●Level 36 marksStart

The number of views,

P

, of a video

t

days after it is uploaded is modelled by

P = ab^t

,

where

a

and

b

are constants.

The graph of

\log_{10} P

against

t

is a straight line with gradient

0.0969

and vertical intercept

2.3010

.(a) Show that this straight-line graph is consistent with the model, explaining how the gradient and intercept relate to

a

and

b

. (2 marks)
(b) Find the values of

a

and

b

, each to 3 significant figures, and interpret the value of

b

in the context of the model. (3 marks)

●●●●●Level 35 marksStart