2.10 Scaling very large and small numbers using logarithms questions for IB Maths AI

31 questions on Scaling very large and small numbers using logarithms, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 2 Functions
HL only

Questions by paper and level

Paper	Questions
Paper 1	16
Paper 2	13
Paper 3	2
Every paper	31

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00390Paper 1 · 6 marks

The energy, $E$ joules, released by an earthquake of magnitude $M$ can be estimated by

\log_{10} E = 4.8 + 1.5 M .

(a)
Find the energy released by an earthquake of magnitude 6.2. Give your answer in the form $a \times 10^{k}$ , where $1 \leq a < 10$ and $k \in Z$ .
2 marks
(b)
Two earthquakes have magnitudes $M_{1}$ and $M_{2}$ , and release energies $E_{1}$ and $E_{2}$ . Show that $\frac{E_{1}}{E_{2}} = 10^{1.5 (M_{1} - M_{2})}$ .
2 marks
(c)
An earthquake of magnitude 7.1 struck a region that had a magnitude 5.9 earthquake a year earlier. Find how many times more energy was released by the larger earthquake.
2 marks

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Question 2 · AI-00301Paper 2 · 15 marks

At an open-air concert in Copenhagen, a loudspeaker emits sound with a power of 0.5 watts. The intensity of the sound, in W m⁻², at a distance of $d$ metres from the speaker is modelled by

I (d) = \frac{0.5}{4 π d^{2}}, d > 0.

The sound level, in decibels (dB), of a sound of intensity $x$ W m⁻² is given by

L (x) = 10 \log_{10} (\frac{x}{10^{- 12}}) .

(a)
Find the intensity of the sound at a distance of 10 m from the speaker. Give your answer in the form $a \times 10^{k}$ , where $1 \leq a < 10$ and $k \in Z$ .
2 marks
(b)
Hence find the sound level at a distance of 10 m from the speaker.
2 marks
(c)
Show that $(L \circ I) (d) = a - 20 \log_{10} d$ , and find the value of $a$ .
4 marks
(d)
Find an expression for $L^{- 1} (x)$ .
3 marks
(e)
Use your answer to part (d) to find the intensity of a sound of 95 dB.
1 mark
(f)
The city council requires the sound level at the nearest house to be no more than 85 dB. Find the minimum distance between the speaker and the nearest house.
3 marks

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Question 3 · AI-01801Paper 3 · 26 marks

This question explores how logarithms can be used to find a power law linking the size of a mammal to its heart rate.

Dr Okafor, a biologist, collected the following data on the mean body mass, $M$ kg, and the mean resting heart rate, $H$ beats per minute, of seven species of mammal.

Species	Mouse	Rat	Rabbit	Dog	Human	Horse	Elephant
$M$ (kg)	0.025	0.3	2.5	20	70	500	5000
$H$ (beats per minute)	620	340	205	108	75	44	30

(a)
Dr Okafor wants to plot the data. Suggest why she should use a logarithmic scale for the mass.
1 mark
(b)
Find the values of $\log_{10} M$ and $\log_{10} H$ for the horse.
2 marks
(c)
Let $X = \log_{10} M$ and $Y = \log_{10} H$ .
Find Pearson's product-moment correlation coefficient, $r$ , for $X$ and $Y$ . Hence comment on whether a model of the form $H = k M^{b}$ is suitable.
3 marks
(d)
Find the equation of the regression line of $Y$ on $X$ .
2 marks
(e)
Use your answer to part (d) to show that $H = k M^{b}$ , and find the value of $k$ and the value of $b$ .
3 marks
(f)
A giraffe has a mass of 800 kg and a measured resting heart rate of 48 beats per minute.
Use your model to estimate the resting heart rate of the giraffe, and find the percentage error in this estimate.
3 marks
(g)
Find the percentage decrease in resting heart rate that this model predicts when body mass is doubled.
2 marks
(h)
Find the mass of a mammal with a predicted resting heart rate of 150 beats per minute.
2 marks
(i)(i)
Dr Okafor models the mean lifespan, $L$ years, of a mammal by $L = 8 M^{0.2}$ . She investigates the total number of heartbeats, $N$ , in a mammal's lifetime, assuming that the heart beats at its resting rate throughout. A year has 525 600 minutes.
Show that $N \approx 1.02 \times 10^{9} \times M^{- 0.058}$ .
3 marks
(i)(ii)
Find the value of $N$ for a mouse and for an elephant. Give your answers in the form $a \times 10^{k}$ , where $1 \leq a < 10$ and $k \in Z$ .
2 marks
(i)(iii)
A popular science article claims that "every mammal gets about one billion heartbeats in its lifetime". Comment on this claim, with reference to your answers to parts (i)(i) and (i)(ii).
2 marks
(j)
A blue whale has a mass of about 150 000 kg. State one reason why Dr Okafor's model may not give a reliable estimate of its heart rate.
1 mark

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Question 4 · AI-00391Paper 1 · 6 marks

The pH of a solution is defined by $pH = - \log_{10} h$ , where $h$ is the concentration of hydrogen ions in moles per litre.

(a)
A sample of lemon juice has a pH of 2.3. Find the concentration of hydrogen ions in the lemon juice.
2 marks
(b)
The concentration of hydrogen ions in a solution is multiplied by a factor $k$ . Show that the pH of the solution decreases by $\log_{10} k$ .
2 marks
(c)
Over ten years, the pH of the water in Lake Vänern fell from 6.8 to 5.9. Find the factor by which the concentration of hydrogen ions increased.
2 marks

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Question 5 · AI-00396Paper 2 · 14 marks

In a laboratory in Nairobi, a technician counts the number of bacteria, $N$ , in a sample $t$ hours after it was prepared.

$t$ (hours)	0	1	2	3	4	5	6
$N$	1200	1900	3300	5100	8800	14 000	23 500

The technician believes that the data can be modelled by $N = k \times a^{t}$ , where $k$ and $a$ are constants.

(a)
Show that $\log_{10} N = t \log_{10} a + \log_{10} k$ .
2 marks
(b)
Hence state what the technician should plot in order to obtain a straight line.
1 mark
(c)
Find the equation of the regression line of $\log_{10} N$ on $t$ .
3 marks
(d)
Hence find the value of $k$ and the value of $a$ .
3 marks
(e)
Use the model to estimate the number of bacteria after 4.5 hours.
2 marks
(f)
Use the model to find the time taken for the number of bacteria to reach one million.
2 marks
(g)
Comment on the reliability of your answer to part (f).
1 mark

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Question 6 · AI-01828Paper 3 · 26 marks

This question explores how logarithms and matrices can be used to locate an animal from the strength of a radio signal.

Batbayar is an ecologist studying snow leopards in the Altai mountains of Mongolia. One leopard wears a collar that sends out a radio signal. Three receiving towers, $T_{1}$ , $T_{2}$ and $T_{3}$ , are at the points $(0, 0)$ , $(10, 0)$ and $(4, 8)$ on a map, where units are kilometres.

The strength of the signal received by a tower, $P$ decibel-milliwatts (dBm), is modelled by

P = - 40 - 20 \log_{10} d,

where $d$ km is the distance from the collar to the tower. At one moment the three towers receive the following signal strengths.

Tower	$T_{1}$	$T_{2}$	$T_{3}$
$P$ (dBm)	−56.5	−54.0	−54.6

(a)
Show that $d = 10^{\frac{- 40 - P}{20}}$ , and hence find the distance from the collar to $T_{1}$ .
3 marks
(b)
Find the distance from the collar to $T_{2}$ and the distance from the collar to $T_{3}$ .
2 marks
(c)
Using the signals at $T_{1}$ and $T_{2}$ only, the collar lies on both of the circles
$x^{2} + y^{2} = d_{1}^{2} and (x - 10)^{2} + y^{2} = d_{2}^{2} .$
Use technology to find the two possible positions of the collar.
3 marks
(d)
Explain how the signal at $T_{3}$ shows which of these positions is correct.
1 mark
(e)
To avoid solving non-linear equations, Batbayar uses the third circle, $(x - 4)^{2} + (y - 8)^{2} = d_{3}^{2}$ , as well.
By subtracting equations, show that
$20 x = 100 + d_{1}^{2} - d_{2}^{2} and 8 x + 16 y = 80 + d_{1}^{2} - d_{3}^{2} .$
2 marks
(f)
The equations in part (e) can be written as $M (\begin{array}{c} x \\ y \end{array}) = b$ .
Write down the matrix $M$ and find the vector $b$ .
2 marks
(g)
Use $M^{- 1}$ to find the position of the collar.
3 marks
(h)
Show that $\det M \neq 0$ . Explain why this method would fail if the three towers were in a straight line.
2 marks
(i)
Batbayar notices that if $d_{1} = d_{2}$ , the first equation in part (e) becomes $x = 5$ .
State the geometrical meaning of the line $x = 5$ in relation to the towers $T_{1}$ and $T_{2}$ , and explain why the collar must lie on it in this case.
2 marks
(j)
Each tower measures signal strength to within 0.5 dBm of the true value.
Find the largest percentage error in the distance from the collar to $T_{2}$ that this could cause.
3 marks
(k)
A GPS reading later shows that the leopard was at $(6, 3)$ at that moment.
Find the distance between this position and the position found in part (g), giving your answer in metres.
2 marks
(l)
Suggest one reason why the model for the signal strength may be less reliable in the mountains.
1 mark

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