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 student investigates how the period $T$ seconds of a pendulum depends on its length $L$ cm. The student believes the relationship can be modelled by

$T = kL^{n}$

where $k$ and $n$ are constants.

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

Find the value of $k$ and the value of $n$ , each to $2$ significant figures, and use the model to estimate, to $3$ significant figures, the period of a pendulum of length $250$ cm. (7 marks)
Level 5 · 7 marksTry it
2
The number of bank branches in a country, $N$ thousand, is modelled by the equation

$N = ab^t$

where $t$ is the time, in years, since the start of $2014$ , and $a$ and $b$ are constants.

The number of branches was recorded at the start of each year from $2014$ to $2022$ . A graph of $\log_{10} N$ against $t$ was drawn for these data, and the line of best fit has equation

$\log_{10} N = 1.06 - 0.032t$
(a) Find the value of $a$ and the value of $b$ , giving each answer to 3 decimal places.

(4 marks)
(b) Interpret, in context,

(i) the value of $a$ ,

(ii) the value of $b$ .

(2 marks)
(c) According to the model, how many bank branches will there be in the country at the start of $2035$ ?

(2 marks)
(d) Comment on how reliable your answer to part (c) is likely to be.

(1 mark)
Level 5 · 9 marksTry it
3
A language-learning app is launched.

The number of active accounts on the app, $N$ thousand, $t$ months after the launch, is modelled by the equation

$N = ab^t$

where $a$ and $b$ are positive constants.
(a) Show that

$\log_{10} N = t\log_{10} b + \log_{10} a$

(2 marks)

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

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

Give your answer to one decimal place.

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

(1 mark)
Level 5 · 10 marksTry it
4
A new indoor climbing centre has opened. The manager suggests that the number of visits to the centre in the $n$ th month after it opened, $V$ , can be modelled by the equation $V = ab^n$ , where $a$ and $b$ are positive constants and $n = 1$ for the first month.

The graph shows the values of $\log_{10} V$ for each of the first eight months, plotted against $n$ .(a) Explain how the graph supports the use of a model of this form.

(3 marks)

The manager suggests the values $a = 400$ and $b = 1.12$ .
(b) Use the graph to show that these values are reasonable.

(2 marks)
(c) Use the model $V = 400 \times 1.12^n$ to estimate the number of visits to the centre in the first month of its second year.

(2 marks)
(d) (i) Use the model $V = 400 \times 1.12^n$ to determine after how many months the total number of visits since the centre opened will first exceed $30\,000$ .

(ii) Explain whether your answer to part (d)(i) is likely to be reliable.

(5 marks)
Level 5 · 12 marksTry it
5
Data relating $y$ and $x$ is believed to follow $y = ax^n$ . A graph of $\log_{10} y$ against $\log_{10} x$ gives a straight line of gradient $1.5$ and intercept $1$ .
(a) Find the values of $a$ and $n$ . (4 marks)
(b) Hence find $y$ when $x = 4$ . (3 marks)
Level 5 · 7 marksTry it

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