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
The variables $x$ and $y$ satisfy the equation $y = Ax^{n}$ , where $A$ and $n$ are constants. The graph of $\ln y$ against $\ln x$ is a straight line passing through the points $(0.500,\ 2.88)$ and $(1.60,\ -0.418)$ , as shown in the diagram.

Find the values of $A$ and $n$ . Give each value correct to the nearest integer.

(4 marks)
Level 3 · 4 marksTry it
2
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 3 · 5 marksTry it
3
A biologist grows a culture of yeast cells. Every hour she records the number of cells in one millilitre of the culture, $N$ million. She proposes the model $N = ab^t$ , where $t$ is the time in hours since the culture was started and $a$ and $b$ are positive constants.

She calculates $\log_{10} N$ for each of her readings and draws a graph of $\log_{10} N$ against $t$ .
(a) Explain why the model implies that the points on this graph should lie close to a straight line.

(3 marks)

The equation of the line of best fit on the graph is $\log_{10} N = 0.68 + 0.125t$ .
(b) Use this equation to estimate the values of $a$ and $b$ , giving each correct to 2 significant figures.

(2 marks)
Level 3 · 5 marksTry it
4
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 3 · 4 marksTry it