A-level · Exponentials & logarithms

Logarithms and non-linear dataExam-style questions

Turning curved laws into straight lines with logs — y = ax^n becomes log-log linear, y = ab^x becomes log-linear — then reading the constants off the gradient and intercept.

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17 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
A mobile phone network records the mean amount of data, $D$ gigabytes, used each month by a smartphone on the network.

The table shows the value of $D$ for some of the years from $2015$ to $2023$
Year $2015$ $2016$ $2018$ $2019$ $2021$ $2022$ $2023$
$D$ (gigabytes) $1.1$ $1.5$ $2.3$ $3.2$ $4.9$ $6.8$ $8.3$
An analyst at the network models $D$ by

$D = a \times b^N$

In this model, $a$ and $b$ are constants, and $N = 0$ in $2015$ , $N = 1$ in $2016$ , and so on.
(a) The analyst plots $\log_{10} D$ against $N$ for the data in the table and draws a line of best fit, shown in the diagram below.The line of best fit has equation

$\log_{10} D = 0.11N + 0.05$
(a) (i) Show that, according to the line of best fit, $a = 1.12$ correct to three significant figures.

(ii) Using the line of best fit, find the value of $b$ to three significant figures.

(2 marks)
(b) State, according to the model, the percentage by which $D$ increases each year.

(1 mark)
(c) (i) Use the model to predict the value of $D$ in $2028$

(ii) Give one reason why the prediction in part (c)(i) may not be reliable.

(3 marks)
Level 4 · 6 marksTry it
2
A motor boat is moving across a calm lake. At time $t = 0$ its engine is switched off.

The speed of the boat, $v$ m s $^{-1}$ , $t$ seconds after the engine is switched off, is modelled by the equation

$v = ab^t$

where $a$ and $b$ are constants.

A graph of $\log_{10} v$ against $t$ is a straight line passing through the points $(0,\ 0.78)$ and $(25,\ 0.28)$
(a) Find, according to the model, the speed of the boat at the instant the engine is switched off.

(2 marks)
(b) Find a complete equation for the model, with $a$ in exact form and $b$ correct to 3 significant figures.

(3 marks)

Exactly $45$ seconds after the engine was switched off, the speed of the boat was measured as $0.74$ m s $^{-1}$
(c) Evaluate the reliability of the model in the light of this measurement.

(2 marks)
Level 4 · 7 marksTry it
3
A new website is launched.

The number of visitors per day to the website, $N$ , $t$ weeks after the launch, is modelled by the equation

$N = ab^t$

where $a$ and $b$ are constants.

A graph of $\log_{10} N$ against $t$ is drawn. The points lie close to a straight line that passes through the points $(2,\ 2.36)$ and $(10,\ 3.24)$
(a) Use these points to find a complete equation for the model, giving the value of $a$ and the value of $b$ to 3 significant figures.

(4 marks)
(b) With reference to the model, interpret the value of the constant $b$ .

(1 mark)

Exactly $20$ weeks after the launch, the website had $12\,400$ visitors in one day.
(c) Use this information to evaluate the reliability of the model.

(2 marks)
Level 4 · 7 marksTry it
4
The number of homes in a town that have a heat pump, $H$ , is counted at the start of each year. A council officer suggests that $H$ can be modelled by the equation $H = ab^t$ , where $t$ is the number of years after the start of 2019 and $a$ and $b$ are positive constants.

The officer plots $\log_{10} H$ against $t$ for the six counts made at the start of the years 2019 to 2024.
(a) Show that, according to the model, $\log_{10} H$ is a linear function of $t$ , and state the gradient and the vertical-axis intercept of its graph in terms of $a$ and $b$ .

(3 marks)

The officer's line of best fit passes through the points $(0, 2.38)$ and $(5, 3.11)$ .
(b) Hence estimate the values of $a$ and $b$ , giving each correct to 2 significant figures.

(2 marks)
(c) Use the model to predict the year during which the number of homes in the town with a heat pump will first exceed $10\,000$ .

(3 marks)
(d) Give one reason why the officer should treat this prediction with caution.

(1 mark)
Level 4 · 9 marksTry it
5
In an experiment, a quantity is modelled by $P = a\,e^{kt}$ . The figure shows the straight-line relationship between $\ln P$ and $t$ . The line has gradient $0.4$ and meets the vertical axis at $1.6$ . Find the values of $a$ and $k$ , giving $a$ to 3 significant figures.
Level 4 · 3 marksTry it