Browse Exponentials & logarithms

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Logarithms

13 questions

LessonNot started
The function f is defined by

f(x)=3x2+13x+4x2+3x−4−2x−1x∈R, x>1\text{f}(x) = \dfrac{3x^2 + 13x + 4}{x^2 + 3x - 4} - \dfrac{2}{x - 1} \qquad x \in \mathbb{R},\ x > 1
(a) Show that f(x)=3x−1x−1\text{f}(x) = \dfrac{3x - 1}{x - 1}

(3 marks)
(b) Show, using calculus, that f is a decreasing function. You must make your reasoning clear.

(3 marks)

The function g is defined by

g(x)=4ln⁡(x+2)−1x⩾−1\text{g}(x) = 4\ln(x + 2) - 1 \qquad x \geqslant -1
(c) Find g−1\text{g}^{-1}

(3 marks)
(d) Find the exact value of aa for which

gf(a)=7\text{gf}(a) = 7

(Solutions relying on calculator technology are not acceptable.)

(4 marks)
●●●●●Level 513 marksStart
The functions f\mathrm{f} and g\mathrm{g} are defined by

f(x)=2∣x−3∣−4,x∈R\mathrm{f}(x) = 2\left|x - 3\right| - 4, \qquad x \in \mathbb{R}

g(x)=ln⁡(x+2),x∈R,  x>−2\mathrm{g}(x) = \ln(x + 2), \qquad x \in \mathbb{R}, \; x > -2
(a) Sketch the graph of y=f(x)y = \mathrm{f}(x), showing the coordinates of the vertex and of any points where the graph meets the coordinate axes, and state the range of f\mathrm{f}. (3 marks)
(b) Solve the equation fg(x)=2\mathrm{fg}(x) = 2, giving your answers in exact form. (4 marks)

The function h\mathrm{h} is defined by

h(x)=2∣x−3∣−4,x⩾k\mathrm{h}(x) = 2\left|x - 3\right| - 4, \qquad x \geqslant k

where kk is a constant, and h−1\mathrm{h}^{-1} exists.
(c) State the least possible value of kk, and find h−1(x)\mathrm{h}^{-1}(x), stating its domain. (3 marks)
●●●●●Level 510 marksStart

The laws of logarithms

45 questions

LessonNot started
Find the general solution of the differential equation

(x3+x2−6x)dydx=(18x−6)y.(x^3 + x^2 - 6x)\frac{dy}{dx} = (18x - 6)y.

Give your answer in the form y=f(x)y = \text{f}(x).

(9 marks)
●●●●●Level 59 marksStart
The gradient of the curve y=f(x)y = \text{f}(x), for x>−2x > -2, is given by the differential equation

2(x+2)dydx=(y−1)(y+3)2(x + 2)\frac{dy}{dx} = (y - 1)(y + 3)

and the curve passes through the point (0,−1)(0, -1). By solving this differential equation show that

f(x)=ax2+bx−8x2+4x+8,\text{f}(x) = \frac{ax^2 + bx - 8}{x^2 + 4x + 8},

where aa and bb are integers to be determined.

(9 marks)
●●●●●Level 59 marksStart
The number of public charging points for electric vehicles in a town, PP, is modelled by the equation

P=abtP = ab^t

where aa and bb are constants and tt is the number of years after the start of 20152015

Using data for the years from 20152015 to 20232023, a graph is plotted of log⁡10P\log_{10} P against tt.

The points lie approximately on a straight line with
  • gradient 0.0860.086
  • intercept 1.741.74 on the log⁡10P\log_{10} P axis
(a) Estimate, to 3 decimal places, the value of aa and the value of bb.

(4 marks)

In the context of the model,
(b) (i) interpret the value of the constant aa,

(ii) interpret the value of the constant bb.

(2 marks)
(c) Use the model to estimate the number of public charging points in the town at the start of 20402040

(2 marks)
(d) Comment on the reliability of the answer to part (c).

(1 mark)
●●●●●Level 59 marksStart
A language-learning app is launched.

The number of active accounts on the app, NN thousand, tt months after the launch, is modelled by the equation

N=abtN = ab^t

where aa and bb are positive constants.
(a) Show that

log⁡10N=tlog⁡10b+log⁡10a\log_{10} N = t\log_{10} b + \log_{10} a

(2 marks)

A graph of log⁡10N\log_{10} N against tt is plotted for the first year after the launch. The points lie close to a straight line that passes through the points (3, 1.20)(3,\ 1.20) and (9, 1.74)(9,\ 1.74)
(b) Hence find a complete equation for the model, giving the value of aa and the value of bb, each to 3 significant figures.

(4 marks)
(c) According to the model, find the value of tt when the number of active accounts reaches 100 000100\,000

Give your answer to one decimal place.

(3 marks)
(d) Give a reason why the model may not be appropriate for large values of tt.

(1 mark)
●●●●●Level 510 marksStart
(a) Use the substitution u=ex+1u = e^x + 1 to show that

∫e3x(ex+1)(ex+5) dx=∫(u−1)2u(u+4) du.\int \dfrac{e^{3x}}{(e^x + 1)(e^x + 5)}\,dx = \int \frac{(u - 1)^2}{u(u + 4)}\,du.

(3 marks)
(b) Hence show that

∫0ln⁡7e3x(ex+1)(ex+5) dx=a+bln⁡2,\int_0^{\ln 7} \dfrac{e^{3x}}{(e^x + 1)(e^x + 5)}\,dx = a + b\ln 2,

where aa and bb are rational numbers to be determined.

(7 marks)
●●●●●Level 510 marksStart
The diagram shows part of the curve y=x1+xy = \dfrac{x}{1 + \sqrt{x}}.14xy(a) Use rectangles of width 0.750.75 to find upper and lower bounds for ∫14x1+x dx\displaystyle\int_1^4 \frac{x}{1 + \sqrt{x}}\,dx. Give your answers correct to 3 significant figures.

(4 marks)
(b) (i) Use the substitution t=xt = \sqrt{x} to show that ∫14x1+x dx=∫122t31+t dt\displaystyle\int_1^4 \frac{x}{1 + \sqrt{x}}\,dx = \displaystyle\int_1^2 \frac{2t^3}{1 + t}\,dt.

(ii) Hence show that ∫14x1+x dx=a+bln⁡32\displaystyle\int_1^4 \frac{x}{1 + \sqrt{x}}\,dx = a + b\ln\tfrac32, where aa and bb are rational numbers to be determined.

(7 marks)
●●●●●Level 511 marksStart
14xyThe diagram shows part of the curve y=2x2+6x+7(x+1)(x+2)y = \dfrac{2x^2 + 6x + 7}{(x + 1)(x + 2)}. The shaded region is bounded by the curve, the xx-axis and the lines x=1x = 1 and x=4x = 4.

Find the exact area of the shaded region, giving your answer in the form p+qln⁡rp + q\ln r, where pp and qq are positive integers and rr is a positive rational number.

(10 marks)
●●●●●Level 510 marksStart
A town council believes that the use of its new cycle-hire scheme can be modelled by H=abtH = ab^t, where aa and bb are both positive constants, HH is the number of hires in a week and tt is the number of complete weeks since the scheme opened.

The numbers of hires in the first six weeks are recorded, and the values of log⁡10H\log_{10} H are plotted against tt in the graph below.1234562.32.42.52.62.72.82.9tlog₁₀ H(a) Explain why the graph suggests that the given model is appropriate.

(3 marks)

The council believes that a=250a = 250 and b=1.2b = 1.2 are good estimates for the parameters in the model.
(b) Show that the graph supports these estimates for the parameters.

(2 marks)
(c) Use the model H=250×1.2tH = 250 \times 1.2^t to predict the number of hires in the tenth week after the scheme opened.

(2 marks)
(d) (i) Use the model H=250×1.2tH = 250 \times 1.2^t to predict the number of weeks after the scheme opened when the total number of hires since opening will first exceed 40 000.

(ii) Comment on how reliable this prediction may be.

(5 marks)
●●●●●Level 512 marksStart
A substance X is decaying exponentially. Its mass, MM grams, at time tt minutes is given by M=120e−0.03tM = 120e^{-0.03t}.
(a) Find the time taken for the mass to decrease to half of its original value.

(3 marks)

A second substance Y is also decaying exponentially. Initially its mass was 90 grams and, after 6 minutes, its mass was 40 grams.
(b) Determine the time at which both substances are decaying at the same rate.

(8 marks)
●●●●●Level 511 marksStart
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The numbers of electric cars and petrol cars registered in a town are being monitored.

When monitoring began, there were 15001500 electric cars registered in the town.

A model predicts that the number of electric cars registered in the town will increase by 16%16\% each year, so that the numbers of electric cars at the end of each year form a geometric sequence.
(a) Find, according to the model, the number of electric cars registered in the town 6 years after monitoring began. Give your answer to 3 significant figures.

(2 marks)

The number of petrol cars registered in the town is monitored over the same period of time.

Given that
  • 3 years after monitoring began there were 16 60016\,600 petrol cars registered in the town
  • 8 years after monitoring began there were 12 20012\,200 petrol cars registered in the town
  • the number of petrol cars registered in the town at the end of each year is modelled as a geometric sequence with equation N=abtN = ab^{t}, where NN is the number of petrol cars tt years after monitoring began and aa and bb are constants
(b) (i) show that b=0.94b = 0.94 to 2 significant figures,

(ii) find the value of aa, giving your answer to 2 significant figures.

(3 marks)

When t=Tt = T, the number of electric cars registered in the town is equal to the number of petrol cars registered in the town.
(c) Using b=0.94b = 0.94 and your value of aa, find, according to the models, the value of TT, giving your answer to one decimal place.

(3 marks)
(d) Find, according to the model for petrol cars, the smallest whole number of years after monitoring began at which there are fewer than 80008000 petrol cars registered in the town.

(3 marks)
●●●●●Level 511 marksStart
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Given that

3x2−6x+9(x+1)(x−2)2≡Ax+1+Bx−2+C(x−2)2\dfrac{3x^2 - 6x + 9}{(x + 1)(x - 2)^2} \equiv \dfrac{A}{x + 1} + \dfrac{B}{x - 2} + \dfrac{C}{(x - 2)^2}
(a) find the values of the constants AA, BB and CC.

(4 marks)
(b) Hence find the exact value of

∫353x2−6x+9(x+1)(x−2)2 dx\int_3^5 \dfrac{3x^2 - 6x + 9}{(x + 1)(x - 2)^2}\,dx

giving your answer in the form pln⁡q+rp\ln q + r where pp, qq and rr are rational numbers.

(6 marks)
●●●●●Level 510 marksStart
A student investigates how the period TT seconds of a pendulum depends on its length LL cm. The student believes the relationship can be modelled by

T=kLnT = kL^{n}

where kk and nn are constants.

The student plots log⁡10T\log_{10} T against log⁡10L\log_{10} L and finds that the points lie close to a straight line passing through (1.20, −0.10)(1.20,\, -0.10) and (1.90, 0.25)(1.90,\, 0.25).

Find the value of kk and the value of nn, each to 22 significant figures, and use the model to estimate, to 33 significant figures, the period of a pendulum of length 250250 cm. (7 marks)
●●●●●Level 57 marksStart
(a) Express 1N(6−N)\dfrac{1}{N(6 - N)} in partial fractions.

(2 marks)

The number of midges near a lake, NN thousand, is modelled by the differential equation

dNdt=N(6−N)sin⁡(t8)12\dfrac{dN}{dt} = \dfrac{N(6 - N)\sin\left(\dfrac{t}{8}\right)}{12}

where tt days is the time after the start of the observations.

Given that, at the start of the observations, there are 20002000 midges near the lake,
(b) solve the differential equation to show that, according to the model,

N=61+2ek(cos⁡(t8)−1)N = \dfrac{6}{1 + 2\text{e}^{k\left(\cos\left(\frac{t}{8}\right) - 1\right)}}

where kk is a constant to be found.

(6 marks)
(c) Hence find, according to the model, the time taken for the number of midges to reach its maximum for the 3rd time.

Give your answer to the nearest day.

(Solutions relying entirely on calculator technology are not acceptable.)

(2 marks)
●●●●●Level 510 marksStart
(a) The equation

x4=e9−2xx^4 = \mathrm{e}^{9 - 2x}

has a single solution, x=αx = \alpha

By considering a suitable change of sign, show that α\alpha lies between 22 and 33

(2 marks)
(b) Show that the equation x4=e9−2xx^4 = \mathrm{e}^{9 - 2x} can be rearranged to give

x=4.5−2ln⁡xx = 4.5 - 2\ln x

(3 marks)
(c) (i) Use the iterative formula

xn+1=4.5−2ln⁡xnx_{n+1} = 4.5 - 2\ln x_n

with x1=3x_1 = 3, to find x2x_2, x3x_3 and x4x_4

Give your answers to three decimal places.

(ii) Figure 1 below shows a sketch of parts of the graphs of y=4.5−2ln⁡xy = 4.5 - 2\ln x and y=xy = x12341234Oy = xy = 4.5 − 2 ln xxyOn Figure 1, draw a staircase or cobweb diagram to show how convergence takes place.

Label, on the xx-axis, the positions of x2x_2, x3x_3 and x4x_4

(iii) Explain why the iterative formula xn+1=4.5−2ln⁡xnx_{n+1} = 4.5 - 2\ln x_n fails to converge to α\alpha when the starting value is x1=e2.25x_1 = \mathrm{e}^{2.25}

(5 marks)
●●●●●Level 510 marksStart

Solving exponential and logarithmic equations

57 questions

LessonNot started
A block of butter is warming so that, tt minutes after it is taken out of a freezer, its temperature is θ\theta °C.

When the butter is taken out of the freezer, its temperature is −18-18 °C. After 2424 minutes its temperature has risen to −6-6 °C.
(a) In a simple model, the rate of increase of the temperature of the butter is assumed to be constant.

(i) Write down a differential equation for this model.

(ii) Solve this differential equation to find θ\theta in terms of tt.

(iii) State one limitation of this model.

(4 marks)
(b) In a revised model, the rate of increase of the temperature of the butter is proportional to the difference between the temperature of the kitchen and the temperature of the butter. The temperature of the kitchen is a constant 2222 °C.

(i) Write down a differential equation for this revised model.

(ii) Solve this differential equation to find θ\theta in terms of tt.

(7 marks)
(c) The butter is soft enough to spread when its temperature is 1515 °C. Find the difference in time between when the two models would predict that the butter is soft enough to spread, giving your answer correct to the nearest minute.

(2 marks)
●●●●●Level 513 marksStart
A language-learning app is launched.

The number of active accounts on the app, NN thousand, tt months after the launch, is modelled by the equation

N=abtN = ab^t

where aa and bb are positive constants.
(a) Show that

log⁡10N=tlog⁡10b+log⁡10a\log_{10} N = t\log_{10} b + \log_{10} a

(2 marks)

A graph of log⁡10N\log_{10} N against tt is plotted for the first year after the launch. The points lie close to a straight line that passes through the points (3, 1.20)(3,\ 1.20) and (9, 1.74)(9,\ 1.74)
(b) Hence find a complete equation for the model, giving the value of aa and the value of bb, each to 3 significant figures.

(4 marks)
(c) According to the model, find the value of tt when the number of active accounts reaches 100 000100\,000

Give your answer to one decimal place.

(3 marks)
(d) Give a reason why the model may not be appropriate for large values of tt.

(1 mark)
●●●●●Level 510 marksStart
A substance X is decaying exponentially. Its mass, MM grams, at time tt minutes is given by M=120e−0.03tM = 120e^{-0.03t}.
(a) Find the time taken for the mass to decrease to half of its original value.

(3 marks)

A second substance Y is also decaying exponentially. Initially its mass was 90 grams and, after 6 minutes, its mass was 40 grams.
(b) Determine the time at which both substances are decaying at the same rate.

(8 marks)
●●●●●Level 511 marksStart
Two new podcasts, AA and BB, are launched on the same day. Each podcast is downloaded 500500 times in Week 1. The number of downloads of podcast AA will increase by 6060 for every subsequent week. The number of downloads of podcast BB will increase by 7%7\% for every subsequent week.
(a) Verify that in Week 10 podcast AA is downloaded 121121 more times than podcast BB, correct to the nearest whole number.

(3 marks)

Let Week XX be the first week in which podcast BB is downloaded more times than podcast AA.
(b) Show that XX satisfies the inequality X>1+log⁡1.07(0.12X+0.88)X > 1 + \log_{1.07}(0.12X + 0.88).

(3 marks)
(c) Use the iterative formula xn+1=1+log⁡1.07(0.12xn+0.88)x_{n+1} = 1 + \log_{1.07}(0.12x_n + 0.88) with x1=20x_1 = 20 to find the value of XX.

You should show the result of each iteration.

(3 marks)
(d) (i) Give a reason why the model for podcast AA may not be realistic.

(ii) Give a different reason why the model for podcast BB may not be realistic.

(2 marks)
●●●●●Level 511 marksStart
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The numbers of electric cars and petrol cars registered in a town are being monitored.

When monitoring began, there were 15001500 electric cars registered in the town.

A model predicts that the number of electric cars registered in the town will increase by 16%16\% each year, so that the numbers of electric cars at the end of each year form a geometric sequence.
(a) Find, according to the model, the number of electric cars registered in the town 6 years after monitoring began. Give your answer to 3 significant figures.

(2 marks)

The number of petrol cars registered in the town is monitored over the same period of time.

Given that
  • 3 years after monitoring began there were 16 60016\,600 petrol cars registered in the town
  • 8 years after monitoring began there were 12 20012\,200 petrol cars registered in the town
  • the number of petrol cars registered in the town at the end of each year is modelled as a geometric sequence with equation N=abtN = ab^{t}, where NN is the number of petrol cars tt years after monitoring began and aa and bb are constants
(b) (i) show that b=0.94b = 0.94 to 2 significant figures,

(ii) find the value of aa, giving your answer to 2 significant figures.

(3 marks)

When t=Tt = T, the number of electric cars registered in the town is equal to the number of petrol cars registered in the town.
(c) Using b=0.94b = 0.94 and your value of aa, find, according to the models, the value of TT, giving your answer to one decimal place.

(3 marks)
(d) Find, according to the model for petrol cars, the smallest whole number of years after monitoring began at which there are fewer than 80008000 petrol cars registered in the town.

(3 marks)
●●●●●Level 511 marksStart
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
(a) Express 10(M+2)(8−M)\dfrac{10}{(M + 2)(8 - M)} in partial fractions.

(2 marks)

A crystal is grown by placing a small seed crystal in a solution.

The mass, MM grams, of the crystal, tt hours after the seed crystal is placed in the solution, is modelled by the differential equation

dMdt=150(M+2)(8−M)0≤M<8, t≥0\frac{dM}{dt} = \frac{1}{50}(M + 2)(8 - M) \qquad 0 \le M < 8,\ t \ge 0

Given that the seed crystal has mass 0.50.5 grams,
(b) solve the differential equation to show that

M=8e0.2t−6e0.2t+3M = \frac{8e^{0.2t} - 6}{e^{0.2t} + 3}

(5 marks)
(c) Find, according to the model, the time taken for the mass of the crystal to reach 66 grams. Give your answer in hours and minutes, to the nearest minute.

(2 marks)
(d) Deduce the maximum mass of the crystal, according to the model.

(1 mark)
●●●●●Level 510 marksStart
(a) Express

200S(40−S)\frac{200}{S(40 - S)}

in partial fractions.

(2 marks)

A gardener is studying the growth of a weed on the surface of a pond.

The area of the surface of the pond covered by the weed, SS m2^2, tt days after the weed was first seen, is modelled by the differential equation

dSdt=1200S(40−S)0<S<40, t⩾0\frac{dS}{dt} = \frac{1}{200}S(40 - S) \qquad 0 < S < 40,\ t \geqslant 0

Given that the weed covered 88 m2^2 of the surface of the pond when it was first seen,
(b) find, according to the model, the time taken for the weed to cover 2020 m2^2 of the surface of the pond. Give your answer in days, to one decimal place.

(5 marks)
(c) Show that

S=AB+e−ktS = \frac{A}{B + e^{-kt}}

where AA, BB and kk are constants to be found.

(3 marks)

As tt increases, the area of the surface of the pond covered by the weed approaches a limiting value LL m2^2.
(d) Find the value of LL, giving a reason for your answer.

(2 marks)
●●●●●Level 512 marksStart
In a chemical reaction two substances combine to form a product. At time tt minutes after the reaction starts, the mass of product formed is xx grams. The reaction is modelled by the differential equation

dxdt=14(2−x)(4−x),0≤x<2\dfrac{\mathrm{d}x}{\mathrm{d}t} = \dfrac{1}{4}(2 - x)(4 - x), \qquad 0 \le x < 2

and x=0x = 0 when t=0t = 0.
(a) Express 1(2−x)(4−x)\dfrac{1}{(2 - x)(4 - x)} in partial fractions. (2 marks)
(b) Hence show that x=4(et/2−1)2et/2−1x = \dfrac{4\left(\mathrm{e}^{t/2} - 1\right)}{2\mathrm{e}^{t/2} - 1}. (5 marks)
(c) Find the value that xx approaches as tt becomes large. Explain, with reference to the differential equation, why the mass of product never reaches this value. (2 marks)
●●●●●Level 59 marksStart
Figure 1dOxhA frog jumps from a point OO on horizontal ground. Figure 1 shows the path of the frog from OO until it lands, a horizontal distance of dd cm from OO.

When the frog has travelled a horizontal distance of xx cm, its height above the ground is hh cm.

The frog is modelled as a particle travelling in a vertical plane above horizontal ground.

The path of the frog is modelled by the equation

h=x−0.25xe0.04x0⩽x⩽dh = x - 0.25x\text{e}^{0.04x} \qquad 0 \leqslant x \leqslant d

Use the model to answer parts (a),
(b) and (c).
(a) Find the value of dd, giving your answer to 2 decimal places.

(Solutions relying entirely on calculator technology are not acceptable.)

(3 marks)
(b) Show that the maximum value of hh occurs when

x=25ln⁡(10025+x)x = 25\ln\left(\dfrac{100}{25 + x}\right)

(4 marks)

Using the iteration formula

xn+1=25ln⁡(10025+xn)x1=15x_{n+1} = 25\ln\left(\dfrac{100}{25 + x_n}\right) \qquad x_1 = 15
(c) (i) find the value of x2x_2 to 2 decimal places,

(ii) find, by repeated iteration, the horizontal distance travelled by the frog before it reaches its maximum height. Give your answer to 2 decimal places.

(3 marks)
●●●●●Level 510 marksStart

Exponential growth and decay models

48 questions

LessonNot started
A block of butter is warming so that, tt minutes after it is taken out of a freezer, its temperature is θ\theta °C.

When the butter is taken out of the freezer, its temperature is −18-18 °C. After 2424 minutes its temperature has risen to −6-6 °C.
(a) In a simple model, the rate of increase of the temperature of the butter is assumed to be constant.

(i) Write down a differential equation for this model.

(ii) Solve this differential equation to find θ\theta in terms of tt.

(iii) State one limitation of this model.

(4 marks)
(b) In a revised model, the rate of increase of the temperature of the butter is proportional to the difference between the temperature of the kitchen and the temperature of the butter. The temperature of the kitchen is a constant 2222 °C.

(i) Write down a differential equation for this revised model.

(ii) Solve this differential equation to find θ\theta in terms of tt.

(7 marks)
(c) The butter is soft enough to spread when its temperature is 1515 °C. Find the difference in time between when the two models would predict that the butter is soft enough to spread, giving your answer correct to the nearest minute.

(2 marks)
●●●●●Level 513 marksStart
A chemist is attempting to model the length of a crystal growing in a solution. The length of the crystal is LL mm at time tt hours after the chemist starts to observe it. When t=0t = 0, L=2L = 2 and when t=1t = 1, L=3L = 3.
(a) The chemist assumes that the rate of increase of the length of the crystal is inversely proportional to the length of the crystal at time tt.

(i) Write down a differential equation to model this situation.

(ii) Solve this differential equation to find LL in terms of tt.

(5 marks)
(b) In a revised model it is assumed that dLdt=ke−0.4tL\dfrac{dL}{dt} = \dfrac{k\text{e}^{-0.4t}}{L}, where kk is a constant. Solve this differential equation to find LL in terms of tt.

(6 marks)
(c) Compare the long-term behaviour of the two models.

(2 marks)
●●●●●Level 513 marksStart
The number of public charging points for electric vehicles in a town, PP, is modelled by the equation

P=abtP = ab^t

where aa and bb are constants and tt is the number of years after the start of 20152015

Using data for the years from 20152015 to 20232023, a graph is plotted of log⁡10P\log_{10} P against tt.

The points lie approximately on a straight line with
  • gradient 0.0860.086
  • intercept 1.741.74 on the log⁡10P\log_{10} P axis
(a) Estimate, to 3 decimal places, the value of aa and the value of bb.

(4 marks)

In the context of the model,
(b) (i) interpret the value of the constant aa,

(ii) interpret the value of the constant bb.

(2 marks)
(c) Use the model to estimate the number of public charging points in the town at the start of 20402040

(2 marks)
(d) Comment on the reliability of the answer to part (c).

(1 mark)
●●●●●Level 59 marksStart
A population is modelled by P=5000e0.08tP = 5000 e^{0.08t}, where tt is the time in years.
(a) State the initial population. (1 mark)
(b) Find the population after 1010 years, to the nearest hundred. (2 marks)
(c) Find, to 3 s.f., the time for the population to reach 1200012000. (3 marks)
(d) Find dPdt\dfrac{\mathrm{d}P}{\mathrm{d}t} when t=10t = 10 and interpret it. (3 marks)
●●●●●Level 59 marksStart
The temperature θ ∘\theta\,^\circC of a cooling drink is modelled by θ=20+60e−0.05t\theta = 20 + 60 e^{-0.05t}, where tt is the time in minutes.
(a) State the initial temperature. (1 mark)
(b) Find the temperature after 1515 minutes, to 1 d.p. (2 marks)
(c) Find the time for the temperature to fall to 40 ∘40\,^\circC, to 3 s.f. (3 marks)
(d) State the temperature the drink approaches in the long term, and why. (2 marks)
●●●●●Level 58 marksStart
A quantity NN is modelled by N=N0ektN = N_0 e^{kt}. When t=0t = 0, N=200N = 200; when t=5t = 5, N=500N = 500.
(a) Write down the value of N0N_0. (1 mark)
(b) Show that k=15ln⁡2.5k = \tfrac15\ln 2.5. (3 marks)
(c) Find the value of NN when t=8t = 8, to the nearest whole number. (3 marks)
●●●●●Level 57 marksStart
The number of bacteria in a culture is modelled by B=300e0.4tB = 300 e^{0.4t}, where tt is the time in hours.
(a) Find the number of bacteria after 66 hours, to the nearest whole number. (2 marks)
(b) Find the rate of increase of BB after 66 hours, to 3 s.f. (3 marks)
(c) Give one reason why this model is unrealistic for large tt. (1 mark)
●●●●●Level 56 marksStart
A town council believes that the use of its new cycle-hire scheme can be modelled by H=abtH = ab^t, where aa and bb are both positive constants, HH is the number of hires in a week and tt is the number of complete weeks since the scheme opened.

The numbers of hires in the first six weeks are recorded, and the values of log⁡10H\log_{10} H are plotted against tt in the graph below.1234562.32.42.52.62.72.82.9tlog₁₀ H(a) Explain why the graph suggests that the given model is appropriate.

(3 marks)

The council believes that a=250a = 250 and b=1.2b = 1.2 are good estimates for the parameters in the model.
(b) Show that the graph supports these estimates for the parameters.

(2 marks)
(c) Use the model H=250×1.2tH = 250 \times 1.2^t to predict the number of hires in the tenth week after the scheme opened.

(2 marks)
(d) (i) Use the model H=250×1.2tH = 250 \times 1.2^t to predict the number of weeks after the scheme opened when the total number of hires since opening will first exceed 40 000.

(ii) Comment on how reliable this prediction may be.

(5 marks)
●●●●●Level 512 marksStart
A substance X is decaying exponentially. Its mass, MM grams, at time tt minutes is given by M=120e−0.03tM = 120e^{-0.03t}.
(a) Find the time taken for the mass to decrease to half of its original value.

(3 marks)

A second substance Y is also decaying exponentially. Initially its mass was 90 grams and, after 6 minutes, its mass was 40 grams.
(b) Determine the time at which both substances are decaying at the same rate.

(8 marks)
●●●●●Level 511 marksStart
tNFigure 1

A group of red squirrels is released into a forest and the number of red squirrels in the forest is then monitored.

The number of red squirrels, NN, in the forest is modelled by the equation

N=180+300e−0.1t−420e−0.3tN = 180 + 300\mathrm{e}^{-0.1t} - 420\mathrm{e}^{-0.3t}

where tt months is the time after the squirrels are released.

Figure 1 is a sketch of NN against tt.

Use the equation of the model to answer parts (a) to (e).
(a) State the number of red squirrels released into the forest.

(1 mark)

In the long term, the number of red squirrels in the forest approaches LL.
(b) State the value of LL.

(1 mark)

The number of red squirrels in the forest reaches its maximum value after TT months.
(c) Find the value of TT, giving your answer to 3 decimal places.

(Solutions based entirely on calculator technology are not acceptable.)

(5 marks)

The number of red squirrels in the forest is 250250 for the second time after MM months.
(d) Show that MM is a solution of the equation

t=10ln⁡(307+42e−0.3t)t = 10\ln\left(\frac{30}{7 + 42\mathrm{e}^{-0.3t}}\right)

(2 marks)

Using the iteration formula

tn+1=10ln⁡(307+42e−0.3tn)with t1=13t_{n+1} = 10\ln\left(\frac{30}{7 + 42\mathrm{e}^{-0.3t_n}}\right) \qquad \text{with } t_1 = 13
(e) (i) find, to 4 decimal places, the value of t2t_2

(ii) find, to 4 decimal places, the value of MM

(3 marks)
●●●●●Level 512 marksStart
A biologist is studying the population of fish in a newly stocked lake. The number of fish, NN thousand, after tt months is modelled by the differential equation

dNdt=kN(8−N)\dfrac{\mathrm{d}N}{\mathrm{d}t} = kN(8 - N)

where kk is a positive constant. When the lake was stocked, t=0t = 0 and N=2N = 2.
(a) Show that N=81+3e−8ktN = \dfrac{8}{1 + 3e^{-8kt}}. (6 marks)
(b) After 1010 months there are 40004000 fish in the lake. Find the exact value of kk. (2 marks)
(c) Find the time taken, according to the model, for the population to reach 60006000 fish. (2 marks)
(d) The biologist claims that, according to the model, the population will eventually reach 80008000 fish.

Use the model to explain whether this claim is correct. (2 marks)
●●●●●Level 512 marksStart
A lake is stocked with fish. The number of fish in the lake, in thousands, tt years after stocking is PP. A model for the population is

dPdt=120P(4−P)\dfrac{\mathrm{d}P}{\mathrm{d}t} = \dfrac{1}{20}P(4 - P)

and P=1P = 1 when t=0t = 0.
(a) Show that, according to the model,

P=41+3e−0.2tP = \dfrac{4}{1 + 3e^{-0.2t}}. (8 marks)
(b) Find the exact value of tt when the model predicts that the population reaches 30003000 fish, giving your answer in the form aln⁡3a\ln 3 where aa is an integer. (2 marks)
(c) State what the model predicts about the fish population in the long term, and give one criticism of the model. (2 marks)
●●●●●Level 512 marksStart
In a chemical reaction, two substances combine to form a product. The mass, mm grams, of product that has formed tt minutes after the reaction starts is modelled by the differential equation

dmdt=k(3−m)(6−m)\dfrac{\mathrm{d}m}{\mathrm{d}t} = k(3 - m)(6 - m), 0≤m<3\qquad 0 \le m < 3,

where kk is a positive constant. When t=0t = 0, m=0m = 0.
(a) Solve the differential equation to show that

m=6(e3kt−1)2e3kt−1m = \dfrac{6\left(e^{3kt} - 1\right)}{2e^{3kt} - 1}. (7 marks)
(b) Given that 11 gram of product has formed after 55 minutes, find the time taken for 22 grams of product to form. Give your answer to the nearest tenth of a minute. (3 marks)
(c) Explain why, according to this model, the mass of product formed can never reach 33 grams. (1 mark)
●●●●●Level 511 marksStart
A biologist is monitoring a colony of seals. At time tt years after monitoring began, the number of seals in the colony is NN hundred.

The biologist models the growth of the colony by the differential equation

dNdt=N(10−N)50\dfrac{\mathrm{d}N}{\mathrm{d}t} = \dfrac{N(10 - N)}{50}

When monitoring began there were 200200 seals in the colony.
(a) Solve the differential equation to show that, according to the model,

N=101+4e−0.2tN = \dfrac{10}{1 + 4\mathrm{e}^{-0.2t}}

(7 marks)
(b) Find, to 1 decimal place, the number of years after monitoring began at which the model predicts the colony will reach 800800 seals. (2 marks)
(c) Explain why, according to this model, the colony can never reach 10001000 seals. (1 mark)
●●●●●Level 510 marksStart
In this question you must show detailed reasoning.

Water flows into a storage tank at a constant rate of 2020 litres per minute. At the same time, water drains from the tank at a rate proportional to the volume currently in the tank. The volume VV litres of water in the tank at time tt minutes satisfies

dVdt=20−kV\dfrac{\mathrm{d}V}{\mathrm{d}t} = 20 - kV,

where kk is a positive constant. The tank is empty at t=0t = 0.
(a) Solve the differential equation to show that V=20k(1−e−kt)V = \dfrac{20}{k}\left(1 - e^{-kt}\right). (7 marks)
(b) Observations over a long period show that the volume of water in the tank approaches, but never reaches, 100100 litres. Write down the value of kk, and hence determine the time taken for the tank to contain 8080 litres, giving your answer in exact form. (4 marks)
(c) The tank has a capacity of 100100 litres. Use the model to explain whether the tank will ever overflow. (1 mark)
●●●●●Level 512 marksStart
A school keeps fish in an aquarium in the shape of a cuboid. The aquarium has a height of 6060 centimetres.

To clean the aquarium, a caretaker drains the water from it using a siphon tube. The end of the tube inside the aquarium is held 88 centimetres above the bottom of the aquarium.

At time tt minutes after the siphon is started, the depth of water, hh centimetres, in the aquarium decreases at a rate which is proportional to h−8h - 8

Initially the aquarium is full, and the depth of water is decreasing at a rate of 1.31.3 centimetres per minute.
(a) Show that

dhdt=−0.025(h−8)\frac{dh}{dt} = -0.025(h - 8)

(3 marks)
(b) Solve the differential equation

dhdt=−0.025(h−8)\frac{dh}{dt} = -0.025(h - 8)

to find an expression for hh in terms of tt

(5 marks)
(c) Find the time taken for the depth of water to fall to 2020 centimetres.

Give your answer to the nearest minute.

(2 marks)
●●●●●Level 510 marksStart
A secondary school has 12001200 students. A new game for mobile phones becomes popular among the students.

When the game is first noticed by the teachers, 1515 students have downloaded it.

The number of students who have downloaded the game is increasing by 20%20\% each day.
(a) The total number of students who have downloaded the game, xx, is modelled by

x=A×Btx = A \times B^t

where AA and BB are constants and tt is the number of days after the game is first noticed.
(a) (i) Use this model to find the total number of students who have downloaded the game 1010 days after it is first noticed.

(ii) Explain why this model is not realistic in the long term.

(5 marks)
(b) A refined model assumes that the rate of increase of the number of students who have downloaded the game is proportional to the product of the number of students who have downloaded the game and the number of students who have not.

When 300300 students have downloaded the game, the number is increasing at a rate of 4545 students per day.
(b) (i) Show that

∫(Ax+B1200−x)dx=∫dt\int \left(\frac{A}{x} + \frac{B}{1200 - x}\right) dx = \int dt

where AA and BB are positive integers to be found.

(ii) Hence, find tt in terms of xx

(iii) Hence, find the number of days it takes from when the game is first noticed until half of the students have downloaded the game.

(10 marks)
●●●●●Level 515 marksStart

Logarithms and non-linear data

20 questions

LessonNot started
Amara is investigating the relationship between the price of a paperback book and the number of copies of it sold each week in a large chain of bookshops.

She takes a random sample of 1515 titles and records, for each title,
  • its price, pp pounds
  • the number of copies sold in one week, qq
The points on Amara's scatter diagram lie close to a curve: as pp increases qq decreases, steeply at first and then more gently.
(a) Interpret the nature of the relationship between qq and pp.

(1 mark)

Amara codes her data using the formulae

x=log⁡10py=log⁡10qx = \log_{10} p \qquad y = \log_{10} q

The product moment correlation coefficient between xx and yy is −0.8463-0.8463

The critical value for a sample of size 1515 at the 5%5\% level of significance is 0.44090.4409
(b) Test whether or not there is significant evidence of a negative correlation between xx and yy.

You should
  • state your hypotheses clearly
  • use a 5%5\% level of significance
  • state the critical value used
(3 marks)

The equation of the line of best fit of yy on xx is

y=3.4−1.6xy = 3.4 - 1.6x
(c) Use the equation of the line of best fit of yy on xx to find a model for qq in terms of pp in the form

q=apkq = ap^{k}

where aa and kk are constants to be found. Give the value of aa to 3 significant figures.

(5 marks)
●●●●●Level 59 marksStart
Femi is investigating how the membership of a new climbing club has grown.

At the end of each of the first 1010 months after the club opened he records
  • the time, tt months, since the club opened
  • the number of members, mm, of the club
The points on Femi's scatter diagram lie close to a curve: as tt increases mm increases, slowly at first and then more and more quickly.
(a) Interpret the nature of the relationship between mm and tt.

(1 mark)

Femi codes his data using the formula

y=log⁡10my = \log_{10} m

The product moment correlation coefficient between tt and yy is 0.93170.9317

The critical value for a sample of size 1010 at the 5%5\% level of significance is 0.54940.5494
(b) Test whether or not there is significant evidence of a positive correlation between tt and yy.

You should
  • state your hypotheses clearly
  • use a 5%5\% level of significance
  • state the critical value used
(3 marks)

The equation of the line of best fit of yy on tt is

y=1.62+0.058ty = 1.62 + 0.058t
(c) Use the equation of the line of best fit of yy on tt to find a model for mm in terms of tt in the form

m=abtm = ab^{t}

where aa and bb are constants to be found. Give the value of aa and the value of bb to 3 significant figures.

(5 marks)
●●●●●Level 59 marksStart
The number of public charging points for electric vehicles in a town, PP, is modelled by the equation

P=abtP = ab^t

where aa and bb are constants and tt is the number of years after the start of 20152015

Using data for the years from 20152015 to 20232023, a graph is plotted of log⁡10P\log_{10} P against tt.

The points lie approximately on a straight line with
  • gradient 0.0860.086
  • intercept 1.741.74 on the log⁡10P\log_{10} P axis
(a) Estimate, to 3 decimal places, the value of aa and the value of bb.

(4 marks)

In the context of the model,
(b) (i) interpret the value of the constant aa,

(ii) interpret the value of the constant bb.

(2 marks)
(c) Use the model to estimate the number of public charging points in the town at the start of 20402040

(2 marks)
(d) Comment on the reliability of the answer to part (c).

(1 mark)
●●●●●Level 59 marksStart
A language-learning app is launched.

The number of active accounts on the app, NN thousand, tt months after the launch, is modelled by the equation

N=abtN = ab^t

where aa and bb are positive constants.
(a) Show that

log⁡10N=tlog⁡10b+log⁡10a\log_{10} N = t\log_{10} b + \log_{10} a

(2 marks)

A graph of log⁡10N\log_{10} N against tt is plotted for the first year after the launch. The points lie close to a straight line that passes through the points (3, 1.20)(3,\ 1.20) and (9, 1.74)(9,\ 1.74)
(b) Hence find a complete equation for the model, giving the value of aa and the value of bb, each to 3 significant figures.

(4 marks)
(c) According to the model, find the value of tt when the number of active accounts reaches 100 000100\,000

Give your answer to one decimal place.

(3 marks)
(d) Give a reason why the model may not be appropriate for large values of tt.

(1 mark)
●●●●●Level 510 marksStart
Data relating yy and xx is believed to follow y=axny = ax^n. A graph of log⁡10y\log_{10} y against log⁡10x\log_{10} x gives a straight line of gradient 1.51.5 and intercept 11.
(a) Find the values of aa and nn. (4 marks)
(b) Hence find yy when x=4x = 4. (3 marks)
●●●●●Level 57 marksStart
Data relating yy and xx is believed to follow y=abxy = ab^x. A graph of log⁡10y\log_{10} y against xx gives a straight line of gradient 0.30.3 and intercept 0.70.7.
(a) Show that log⁡10y=log⁡10a+xlog⁡10b\log_{10} y = \log_{10} a + x\log_{10} b. (2 marks)
(b) Find the values of aa and bb, to 3 s.f. (4 marks)
●●●●●Level 56 marksStart
The table suggests y=abxy = ab^x. A graph of log⁡10y\log_{10} y against xx is drawn and is a straight line passing through (0,0.5)(0, 0.5) and (4,2.1)(4, 2.1).
(a) Find the gradient of the line, and explain what it represents. (3 marks)
(b) Find the values of aa and bb, to 3 s.f. (4 marks)
●●●●●Level 57 marksStart
A town council believes that the use of its new cycle-hire scheme can be modelled by H=abtH = ab^t, where aa and bb are both positive constants, HH is the number of hires in a week and tt is the number of complete weeks since the scheme opened.

The numbers of hires in the first six weeks are recorded, and the values of log⁡10H\log_{10} H are plotted against tt in the graph below.1234562.32.42.52.62.72.82.9tlog₁₀ H(a) Explain why the graph suggests that the given model is appropriate.

(3 marks)

The council believes that a=250a = 250 and b=1.2b = 1.2 are good estimates for the parameters in the model.
(b) Show that the graph supports these estimates for the parameters.

(2 marks)
(c) Use the model H=250×1.2tH = 250 \times 1.2^t to predict the number of hires in the tenth week after the scheme opened.

(2 marks)
(d) (i) Use the model H=250×1.2tH = 250 \times 1.2^t to predict the number of weeks after the scheme opened when the total number of hires since opening will first exceed 40 000.

(ii) Comment on how reliable this prediction may be.

(5 marks)
●●●●●Level 512 marksStart
A student investigates how the period TT seconds of a pendulum depends on its length LL cm. The student believes the relationship can be modelled by

T=kLnT = kL^{n}

where kk and nn are constants.

The student plots log⁡10T\log_{10} T against log⁡10L\log_{10} L and finds that the points lie close to a straight line passing through (1.20, −0.10)(1.20,\, -0.10) and (1.90, 0.25)(1.90,\, 0.25).

Find the value of kk and the value of nn, each to 22 significant figures, and use the model to estimate, to 33 significant figures, the period of a pendulum of length 250250 cm. (7 marks)
●●●●●Level 57 marksStart