Logarithms — Free IB MYP4 Mathematics (Extended) Questions

1QuestionBasic log function graphsConcept Practice

2 marks~3 minCriterion A

The diagram shows the graph of

y = \log_2 x

for

0 < x \leq 8

.

a

State the coordinates of the point where the graph crosses the

x

-axis. [1]

b

State the equation of the vertical asymptote of the graph. [1]

Solutions

2QuestionEvaluating simple logarithmsAssessment Practice

7 marks~11 minCriterion B

The diagram shows the first three figures of a pattern made from small squares arranged in an L-shape.

Figure 1 has 3 squares, Figure 2 has 8 squares, and Figure 3 has 15 squares.

a

Write down the number of squares in Figure 4. [1]

b

Find a rule for the total number of squares

S

in Figure

n

, giving your answer in the form

S = an^2 + bn

, where

a

and

b

are integers to be found. [3]

c

A student claims that Figure

n

always contains an odd number of squares. Justify whether this claim is correct for all positive integers

n

, referring to the structure of your rule. [3]

Solutions

3QuestionEvaluating simple logarithmsAssessment Practice

10 marks~15 minCriterion A

The diagram shows the graph of

f(x) = \log_2 x

for

0 < x \leq 16

, with three points

A

,

B

, and

C

marked on the curve at

x = 1

,

x = 4

, and

x = 16

respectively.

a

Find the coordinates of points

A

,

B

, and

C

, giving exact values. [3]

b

A student claims that the point

P\!\left(\tfrac{1}{8},\, k\right)

lies on the graph of

g(x) = \log_2 x + \log_2(4x)

. Find the value of

k

, showing all steps of your reasoning. [3]

c

The equation

\log_2(x+3) + \log_2(x-1) = 5

has exactly one valid solution. Solve the equation and justify why the other algebraic solution must be rejected. [4]

Solutions

4QuestionEvaluating simple logarithmsAssessment Practice

12 marks~18 minCriterion C

Two major earthquakes are compared using the Richter scale formula

M = \log_{10}\!\left(\frac{I}{I_0}\right),

where

M

is the magnitude,

I

is the intensity of the earthquake, and

I_0

is a fixed reference intensity.

The 1906 San Francisco earthquake had magnitude

M = 7.8

, the 2011 Tohoku earthquake had magnitude

M = 9.1

, and the 1960 Valdivia earthquake had magnitude

M = 8.6

.

The diagram shows the three magnitudes plotted on a number line.

a

Calculate how many times more intense the Tohoku earthquake was compared to the San Francisco earthquake. Give your answer to 3 significant figures. [3]

b

A seismologist claims that the Valdivia earthquake released approximately 8 times more energy than the San Francisco earthquake, using the empirical relationship

\log_{10}(E) = 4.8 + 1.5M,

where

E

is the energy released in joules. Show that this claim is incorrect by calculating the ratio of energies, and find the correct ratio to 3 significant figures. [4]

c

The table below shows the Richter magnitude and the energy ratio relative to the San Francisco earthquake (

M = 7.8

) for three earthquakes.

Earthquake — Magnitude — Energy ratio (relative to San Francisco)

San Francisco	7.8	1.00
Valdivia	8.6	$R$
Tohoku	9.1	$S$

Using your results from parts (a) and (b), or otherwise, assess whether the Richter magnitude scale gives an accurate impression of the relative destructive potential of these earthquakes. Justify your assessment by referring to both the intensity ratios and the energy ratios in your answer. [5]

Solutions

5QuestionNatural logarithms (base e)Assessment Practice

14 marks~21 minCriterion D

An investor places 5000 dollars into a savings account. The bank offers two options:

Option A: 4.8% annual interest compounded monthly.
Option B: 4.75% annual interest compounded quarterly.

The balance after

t

years under Option A is modelled by

A(t) = 5000\left(1 + \frac{0.048}{12}\right)^{12t},

and under Option B by

B(t) = 5000\left(1 + \frac{0.0475}{4}\right)^{4t}.

The table below shows the actual recorded balances at selected years.

Year ( $t$ )	5	10	20
Actual balance (dollars)	6283	7891	12530

a

Calculate the balance predicted by Option A at

t = 5

,

t = 10

, and

t = 20

. Give each answer to the nearest dollar. [3]

b

The investor states: "Option A always gives a higher balance than Option B." Using your results from part (a) and calculations for Option B at

t = 5

,

t = 10

, and

t = 20

, determine whether this statement is correct. Give all balances to the nearest dollar. [4]

c

The percentage error of a model prediction is given by

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%.

Using Option A predictions from part (a) and the actual balances in the table, calculate the percentage error at each of

t = 5

,

t = 10

, and

t = 20

. Give each answer to 2 decimal places. [3]

d

The investor is deciding whether to use Option A as a long-term financial plan. Using the percentage errors found in part (c), advise the investor on whether Option A is a reliable model for their actual balance over time, and suggest one reason why the model may not reflect the actual balance in the long term. [4]

Solutions

6QuestionQuotient ruleAssessment Practice

6 marks~9 minCriterion B

The diagram shows a growing pattern of tiles arranged in L-shaped figures. Figure 1 has 3 tiles, Figure 2 has 5 tiles, and Figure 3 has 7 tiles.

a

Write down the number of tiles in Figure 4. [1]

b

Find a rule for the number of tiles

T

in Figure

n

, giving your answer in the form

T = an + b

, where

a

and

b

are integers. [2]

c

The total number of tiles used to build Figures 1 through

n

is given by

n^2 + 2n

. Verify this formula for

n = 3

, then justify whether this formula confirms that the cumulative total grows faster than the per-figure tile count as

n

increases. [3]

Solutions

7QuestionQuotient ruleAssessment Practice

10 marks~15 minCriterion A

A company models its monthly profit

P

(in thousands of dollars) using the function

P(x) = \log_2(4x) + \log_2\!\left(\frac{x}{2}\right)

where

x

is the number of units sold (in hundreds) and

x > 0

.

a

Show that

P(x)

can be written in the form

P(x) = 2\log_2 x + k

, where

k

is an integer to be found. [3]

b

The table below shows the model's predicted profit alongside observed data at three production levels.

x

(hundreds of units): 1, 2, 4

Predicted

P

(thousands of dollars): 1, 3, 5

Actual

P

(thousands of dollars): 0.8, 2.6, 5.3

Calculate the percentage error for each value of

x

, giving your answers to 2 decimal places, using

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%

and hence determine at which production level the model is least reliable. [4]

c

The company sets a profit target of

P(x) = 5

. Using your result from part (a), solve for the exact value of

x

. Justify whether the company should rely on this model to meet the target, given that the maximum production capacity is 400 units per month and the model's reliability varies across production levels. [3]

Solutions

8QuestionQuotient ruleAssessment Practice

12 marks~18 minCriterion D

An environmental scientist monitors seismic activity in a region prone to earthquakes. The Richter scale assigns a magnitude

M

to each earthquake, where the intensity

I

of an earthquake of magnitude

M

is given by

I = I_0 \times 10^{M},

and

I_0

is a fixed reference intensity. The magnitude difference between two earthquakes satisfies

\Delta M = \log_{10}\!\left(\frac{I_1}{I_2}\right).

The scientist records four earthquakes. The predicted intensity ratio is measured relative to a reference earthquake of magnitude 4.0.

Earthquake A — Magnitude: 4.0 — Predicted ratio: 1.00

Earthquake B — Magnitude: 4.5 — Predicted ratio: ?

Earthquake C — Magnitude: 5.2 — Predicted ratio: ? — Observed ratio: 158

Earthquake D — Magnitude: 6.1 — Predicted ratio: ?

a

Calculate the predicted intensity ratio for earthquakes B, C, and D, giving each answer to 3 significant figures. [3]

b

Using the formula

\text{percentage error} = \frac{|\text{predicted} - \text{observed}|}{\text{observed}} \times 100\%,

calculate the percentage error for earthquake C. Give your answer to 1 decimal place. [2]

c

A building code requires that any earthquake whose intensity is at least 200 times the reference intensity

I_0 \times 10^{4.0}

must trigger an automatic structural inspection. Deduce the minimum magnitude

M

that triggers an inspection, giving your answer to 2 decimal places. [3]

d

The scientist claims that the Richter scale is a reliable model for communicating earthquake severity to the public because equal magnitude differences always correspond to equal intensity ratios. Assess this claim, and justify whether the Richter scale is an appropriate communication tool for non-specialist audiences. [4]

Solutions

9QuestionExpanding logarithmic expressionsAssessment Practice

12 marks~18 minCriterion C

A seismologist models earthquake intensity using the Richter scale, where the magnitude

M

of an earthquake is defined by

M = \log_{10}\!\left(\frac{I}{I_0}\right),

where

I

is the intensity of the earthquake and

I_0

is a fixed reference intensity.

Two earthquakes are recorded at the same monitoring station. Earthquake A has intensity

I_A = 10^{6}\,I_0

and Earthquake B has intensity

I_B = 10^{9}\,I_0

. The energy

E

released by an earthquake satisfies

E \propto I^{3/2}

.

Give all numerical answers to 3 significant figures where appropriate.

a

Find the Richter magnitude of each earthquake, showing your use of the formula. [2]

b

Show that

M_B - M_A = 3

, and hence deduce the ratio

\dfrac{I_B}{I_A}

. [3]

c

Calculate how many times more energy Earthquake B releases compared to Earthquake A, giving your answer in the form

10^k

where

k

is an exact fraction. [3]

d

A journalist reports that Earthquake B is "three times more destructive" than Earthquake A because its magnitude is 3 units higher. Critique this claim, justifying your answer with reference to both the intensity ratio and the energy ratio found in parts (b) and (c). [4]

Solutions

10QuestionConverting logarithmic equations to exponential formAssessment Practice

5 marks~8 minCriterion C

The graph of

f(x) = \log_2 x

is shown in the diagram for

x > 0

.

a

Write down the value of

f(8)

. [1]

b

Find the value of

x

for which

f(x) = -2

, giving your answer as a fraction. [2]

c

A second function is defined as

g(x) = \log_2(x - 3)

. Justify whether a student who claims the vertical asymptote of

g(x)

is

x = 0

and the domain is

x > 0

is correct. [2]

Solutions

11QuestionConverting exponential equations to logarithmic formAssessment Practice

7 marks~11 minCriterion A

The diagram shows three stages of a pattern made from small squares arranged in an L-shape.

Stage 1 has 3 squares, Stage 2 has 5 squares, and Stage 3 has 7 squares.

a

Write down the number of squares in Stage 4 and Stage 5. [2]

b

Find a formula for

S

, the number of squares in Stage

n

, where

n

is a positive integer. Show your working clearly. [2]

c

A student claims that no stage of this pattern can contain a perfect square number of squares. Justify whether this claim is correct. [3]

Solutions

12QuestionConverting exponential equations to logarithmic formAssessment Practice

6 marks~9 minCriterion D

A sound engineer uses the formula

L = 10\log_{10}\!\left(\frac{I}{I_0}\right)

to model the loudness

L

(in decibels, dB) of a sound, where

I

is the sound intensity and

I_0

is the reference intensity.

The engineer measures three sounds and records the following data.

Intensity ratio

I/I_0

: 100 | 1000 | 50 000

Predicted

L

(dB): (i) | (ii) | (iii)

Actual

L

(dB): 21 | 32 | 49

(a) Calculate the three predicted values of

L

. Give each answer to the nearest whole number. [2]

(b) Find the percentage error for each sound using

\text{percentage error} = \frac{|\,\text{predicted} - \text{actual}\,|}{\text{actual}} \times 100\%.

Give each answer to 1 decimal place. [2]

(c) The engineer claims the model is reliable across all three sounds. Justify whether this claim is supported by your results from parts (a) and (b), and identify one limitation of applying this model at very high intensity ratios. [2]

Solutions

13QuestionConverting exponential equations to logarithmic formAssessment Practice

7 marks~11 minCriterion B

The diagrams below show Figures 1, 2, 3, and 4 of a pattern built from unit squares arranged in an L-shape.

The number of unit squares in each figure is recorded below.

Figure number ( $n$ )	1	2	3	4
Number of squares ( $S$ )	3	7	13	21

a

Write down the number of unit squares in Figure 5. [1]

b

Find a quadratic rule for

S

in terms of

n

, giving your rule in the form

S = an^2 + bn + c.

Show all working. [3]

c

A student claims that Figure 50 contains more than 2600 unit squares. Justify whether this claim is correct. [3]

Solutions

14QuestionSolving equations using log lawsAssessment Practice

11 marks~17 minCriterion C

The table below shows values of four logarithmic expressions for

x = 1, 2, 4, 8

.

$x$	1	2	4	8
$\log_2 x$	0	1	2	3
$\log_2(2x)$	1	2	3	4
$\log_2(x^2)$	0	2	4	6
$\log_2(\sqrt{x})$	0	0.5	1	1.5

a

Using the table, find the values of

k

,

p

, and

q

such that, for all

x > 0

,

\log_2(2x) = \log_2 x + k, \qquad \log_2(x^2) = p\log_2 x, \qquad \log_2(\sqrt{x}) = q\log_2 x.

For each value, show how the table supports your answer. [3]

b

A student claims that

\log_2(2x) = \log_2(x^2)

whenever

x = 2

. Using your rules from part (a), find all positive values of

x

for which

\log_2(2x) = \log_2(x^2)

, and verify your answer by substituting back into both expressions. [4]

c

The student conjectures that all three rules from part (a) follow from the single law

\log_b(mn) = \log_b m + \log_b n

. Justify whether this conjecture is correct by deriving each rule directly from that law, without using the table. [4]

Solutions

15QuestionSolving equations using log lawsAssessment Practice

7 marks~11 minCriterion B

The diagram shows the first three figures of a pattern of nested squares.

Figure 1 is a single square with side length

2

cm.
Figure 2 is formed by placing a square of side length

4

cm around Figure 1, leaving a border of width

1

cm on each side.
Figure 3 is formed by placing a square of side length

6

cm around Figure 2, leaving a border of width

1

cm on each side.

The area

A_n

(in cm²) of the outermost square ring added at Figure

n

follows the pattern:

A_1 = 2^2 = 4, \quad A_2 = 4^2 - 2^2 = 12, \quad A_3 = 6^2 - 4^2 = 20

a

Write down

A_4

, the area of the outermost ring added in Figure 4. [1]

b

Find a rule for

A_n

in the form

A_n = an + b

, where

a

and

b

are integers. [3]

c

Justify why

A_n

must be a linear function of

n

by referring to the structure of the squares in the pattern. [3]

Solutions

16QuestionChecking for extraneous solutionsAssessment Practice

10 marks~15 minCriterion D

A seismologist uses the Gutenberg–Richter energy model

\log_{10}(E) = 4.4 + 1.5M

where

E

is the energy released in joules and

M

is the Richter magnitude of an earthquake.

Three earthquakes were recorded in a seismically active region.

Earthquake	A	B	C
Magnitude $M$	5.0	7.0	9.0
Predicted $E$ (J)	?	?	?
Measured $E$ (J)	$3.2 \times 10^{9}$	$3.5 \times 10^{12}$	$8.0 \times 10^{17}$

a

Calculate the predicted energy

E

(in joules, to 3 significant figures) for each earthquake and complete the table. [3]

b

Calculate the percentage error between the predicted and measured energy for earthquake

C

, giving your answer to 1 decimal place. Use

\text{percentage error} = \frac{|\,\text{predicted} - \text{measured}\,|}{\text{measured}} \times 100\%.

[2]

c

A new earthquake releases a measured energy of

E = 5.0 \times 10^{16}

J. Using the model, determine the magnitude

M

predicted for this earthquake, giving your answer to 2 decimal places. [2]

d

The diagram shows the model curve together with the three measured data points from the table. Using your results from parts (a) and (b), assess whether the seismologist should rely on this model to predict earthquake energy across all three magnitude levels, and identify one limitation that affects its reliability. [3]

Solutions

17QuestionSolving equations with different basesAssessment Practice

9 marks~14 minCriterion A

The diagram shows the graphs of

f(x) = \log_2(x)

and

g(x) = \log_3(x)

for

0 < x \leq 32

, together with the horizontal line

y = 5

.

a

Using the change-of-base formula, show that the equation

\log_2(x) + \log_3(x) = 5

can be written as

\log_{10}(x) = \frac{5\,\log_{10}(2)\,\log_{10}(3)}{\log_{10}(2) + \log_{10}(3)}.

[3]

b

Hence find the value of

x

that satisfies

\log_2(x) + \log_3(x) = 5

. Give your answer correct to 3 significant figures. [2]

c

A student claims that the equation

\log_2(x) + \log_3(x) = k

has a solution for every real number

k

. Justify whether this claim is correct, and determine the value of

k

for which the solution is

x = 6

, giving

k

in exact form using logarithms. [4]

Solutions

18QuestionSolving equations with unknown exponentsAssessment Practice

11 marks~17 minCriterion A

A scientist models the temperature

T

(in °C) of a cooling liquid at time

t

minutes using the equation

T(t) = 80 \cdot 3^{-kt},

where

k

is a positive constant.

Observed and predicted temperatures at three values of

t

are shown below.

$t$ (min)	5	10	20
Observed $T$ (°C)	38.0	18.5	4.3
Predicted $T$ (°C)	?	18.5	?

The value

k

is determined using the observed temperature at

t = 10

minutes.

a

Using

T = 18.5

when

t = 10

, show that

k = \frac{1}{10}\log_{3}\!\left(\frac{80}{18.5}\right),

and hence find the value of

k

correct to 3 significant figures. [4]

b

Using your value of

k

from part (a), find the model's predicted temperatures at

t = 5

and

t = 20

, each correct to 1 decimal place. Hence calculate the percentage error for each prediction using

\text{percentage error} = \frac{|\,\text{predicted} - \text{observed}\,|}{\text{observed}} \times 100\%,

giving each percentage error correct to 1 decimal place. [4]

c

The scientist claims the model is reliable for

0 \leq t \leq 20

. Justify this claim, or reject it, by referring to the percentage errors found in part (b), and state one limitation of using this exponential model to predict the temperature for large values of

t

. [3]

Solutions

19QuestionApplying logarithms to both sidesAssessment Practice

10 marks~15 minCriterion D

A colony of bacteria grows so that its population

P

at time

t

hours satisfies

P(t) = P_0 \cdot 2^{t/k},

where

P_0

is the initial population and

k

is a positive constant.

A scientist records the following data.

Time $t$ (hours)	0	3	9
Population $P$	2000	2828	5657

The scientist uses the model

P(t) = 2000 \cdot 2^{t/6}

to predict future population values.

a

Show that

k = 6

is consistent with the data point at

t = 3

, giving your answer to 4 significant figures. [3]

b

The scientist claims the colony will reach a population of

3^x

at some time

t

, where

x

satisfies

2^x = 3^{x-1}.

Solve this equation for

x

, giving your answer in exact form and as a decimal to 3 significant figures. [4]

c

A second scientist argues that the model

P(t) = 2000 \cdot 2^{t/6}

will eventually overestimate the true population because real bacterial colonies face resource constraints. Assess the validity of this argument by stating one assumption of the model and one limitation that arises from it. [3]

Solutions

20QuestionApplying logarithms to both sidesAssessment Practice

7 marks~11 minCriterion B

The diagram shows the first four figures of a pattern made from small squares arranged in an L-shape.

The table below shows the number of small squares

S

in each figure.

Figure number ( $n$ )	1	2	3	4
Number of squares ( $S$ )	3	5	7	9

a

Write down the number of small squares in Figure 5. [1]

b

Find a rule for the number of small squares

S

in Figure

n

, giving your answer in the form

S = an + b

where

a

and

b

are integers to be determined. [2]

c

A large version of this pattern is built using exactly 47 small squares. Determine which figure number this is, showing your working. [1]

d

Justify why the rule

S = an + b

can never produce a value of

S

that is an even number, for any positive integer

n

. [3]

Solutions

21QuestionSolving equations with unknown exponentsAssessment Practice

10 marks~15 minCriterion C

The diagram shows the graphs of

f(x) = 5^x

and

g(x) = 2^{x+3}

for

-1 \leq x \leq 5

.

a

Show that the equation

5^x = 2^{x+3}

can be written as

x = \frac{3\log 2}{\log 5 - \log 2}.

[3]

b

Hence find the coordinates of the point of intersection of the two graphs, giving the

x

-value in exact form and the

y

-value to 3 significant figures. [3]

c

A student claims that for any positive constant

k

, the equation

5^x = k \cdot 2^{x+3}

always has exactly one solution. Justify whether this claim is correct, giving the solution in exact form in terms of

k

and

\log

, and state what happens to

x

as

k \to 0^+

. [4]

Solutions

22QuestionVertical asymptotesAssessment Practice

7 marks~11 minCriterion D

A chemist uses the function

f(x) = \log_{10}(x - 2)

to model the pH of a solution, where

x

is the total volume of acid added, in mL, and

x > 2

.

The diagram shows the graph of

f

for

2 < x \leq 12

, together with three observed pH readings recorded in the laboratory.

Observed data:

x

(mL): 4, 7, 12

Observed pH:

-0.52

,

0.70

,

1.00

a

Write down the equation of the vertical asymptote of

f

and interpret its meaning in the context of the acid volume. [2]

b

Complete the table of predicted pH values, giving each answer to 2 decimal places.

x

(mL): 4, 7, 12

Predicted pH: ?, ?, ?

Hence calculate the percentage error at

x = 7

, using

\text{percentage error} = \frac{|\text{predicted} - \text{observed}|}{|\text{observed}|} \times 100\%.

Give your answer to 1 decimal place. [3]

c

The chemist considers using the model to predict the pH when

x = 2.05

mL. Justify whether the model is reliable for volumes this close to 2 mL, using the values of

f(2.05)

and

f(2.5)

in your reasoning. [2]

Solutions

23QuestionTransformations (shifts, reflections, stretches)Assessment Practice

10 marks~15 minCriterion C

The graph of

f(x) = \log_2 x

is transformed to obtain the graph of

g(x)

. The transformation consists of a horizontal translation of 3 units in the positive

x

-direction, followed by a vertical stretch with scale factor 4 from the

x

-axis.

a

Find the equation of

g(x)

and state its domain. [3]

b

The graph of

g(x)

passes through the point

(a, 12)

. Find the exact value of

a

. [3]

c

A student claims that

g(x) = f(x)

has no solution because the two graphs never intersect. Justify whether the student is correct, showing all algebraic reasoning. [4]

Solutions

24QuestionComparing exponential and logarithmic graphsAssessment Practice

6 marks~9 minCriterion B

The diagram shows the first four figures of a pattern made from small squares arranged in two rows.

Figure 1 has 2 squares, Figure 2 has 6 squares, Figure 3 has 12 squares, and Figure 4 has 20 squares.

a

Write down the number of squares in Figure 5. [1]

b

Find a rule for the number of squares

S

in Figure

n

in the form

S = an^2 + bn

, where

a

and

b

are integers. [2]

c

A student claims that Figure

n

always has exactly

n

more squares than Figure

n - 1

. Justify whether this claim is correct, showing your working clearly. [3]

Solutions

25QuestionRichter scale (earthquakes)Assessment Practice

8 marks~12 minCriterion A

The table below shows the magnitude

M

and amplitude ratio

A

(relative to a reference earthquake of magnitude 0) for four earthquakes recorded at a seismology station.

Magnitude

M

: 1.0, 2.0, 3.0, 4.0

Amplitude ratio

A

: 10, 100, 1000, 10 000

The relationship between

M

and

A

follows the model

A = 10^M

.

a

Determine the value of

M

when

A = 5000

, giving your answer to 3 significant figures. [3]

b

A second station records an earthquake with amplitude ratio

A = 10^{3.7}

. A third station, at a different location, records the same earthquake and measures an amplitude ratio 4 times greater. Find the magnitude recorded by the third station, giving your answer to 3 significant figures. [3]

c

A seismologist claims that an earthquake of magnitude 6.0 has an amplitude ratio exactly 1000 times greater than one of magnitude 3.0. Justify whether this claim is correct. [2]

Solutions

26QuestionSolving real-life exponential problemsAssessment Practice

9 marks~14 minCriterion C

A sample of 100 grams of a radioactive isotope decays by half every 8 days. The table below shows the mass remaining after each half-life period.

Number of half-lives ( $n$ )	0	1	2	3	4
Mass remaining (g)	100	50	25	12.5	6.25

a

Find the mass remaining after 7 half-lives. Give your answer in grams. [2]

b

The mass

M

grams after

n

half-lives can be written in the form

M = 100 \times k^n.

Show that

k = 0.5

, and hence find the number of complete half-lives required for the mass to fall below 1 gram for the first time. [4]

c

A student claims that after 20 half-lives the mass remaining is exactly 0 grams, because "the sample keeps halving and must eventually reach zero." Justify whether this claim is mathematically correct, referring to the structure of the exponential model. [3]

Solutions

27QuestionSolving real-life exponential problemsAssessment Practice

8 marks~12 minCriterion D

A patient receives a 200 mg dose of a medication that decays exponentially. The amount

A

(in mg) remaining after

t

hours is modelled by

A(t) = 200(0.85)^t, \quad t \geq 0.

The diagram shows the graph of

A(t)

for

0 \leq t \leq 15

, together with three observed measurements from the patient's blood tests.

$t$ (hours)	2	6	10
Predicted $A$ (mg)	144.5	75.4	39.4
Observed $A$ (mg)	138	80	46

The doctor requires the drug concentration to fall below 50 mg before administering the next dose.

a

Show that the predicted time at which the drug concentration first falls below 50 mg is

t = 9

hours. [3]

b

Find the percentage error between the predicted and observed concentrations at

t = 6

hours, giving your answer to one decimal place. Hence comment on whether the model's prediction at

t = 10

hours is reliable enough to use as the basis for the dosing decision, referring to the errors in your table. [3]

c

State one assumption of this exponential decay model. Advise the doctor whether the predicted dosing time of

t = 9

hours should be used to make the dosing decision, justifying your answer by referring to the model's limitations outside

0 \leq t \leq 10

. [2]

Solutions

28QuestionGrowth and decay problemsAssessment Practice

10 marks~15 minCriterion B

The diagram shows the first four figures in a sequence built by adding a new border of squares around the previous figure.

Figure $n$	1	2	3	4
Total squares $S$	1	5	13	25

a

Complete the table below by finding the number of squares added when going from Figure

n

to Figure

n+1

, for

n = 1, 2, 3

. Describe the pattern in these differences. [3]

Squares added: \_\_\_, \_\_\_, \_\_\_

b

Show that the total number of squares in Figure

n

is given by

S = 2n^2 - 2n + 1.

[4]

c

A student claims that Figure 20 will have more than 750 squares. Justify whether this claim is correct, and find the smallest value of

n

for which

S > 750

. [3]

Solutions

29QuestionConverting between basesAssessment Practice

13 marks~20 minCriterion D

An acoustic engineer tests two sound sources using the formula

L = 10 \log_{10}\!\left(\frac{I}{I_0}\right)

where

I_0 = 10^{-12}

W/m² is the standard reference intensity,

I

is the sound intensity in W/m², and

L

is the sound level in decibels (dB). The jet engine measures

L_j = 120

dB and the laboratory fan measures

L_f = 42

dB.

a

Find the intensities

I_j

and

I_f

in W/m². Give your answers in exact form. [3]

b

The engineer proposes a rescaled model using a new reference intensity

I_0' = 10^{-10}

W/m², so that a barely audible whisper in the lab registers 0 dB. Show that on this rescaled model the jet engine registers exactly 100 dB and the fan registers exactly 22 dB. [3]

c

The table below compares the rescaled model levels with readings from an independent calibrated meter for three sound sources.

Sound source: Jet engine | Fan | Door slam

Rescaled model (dB): 100 | 22 | 65

Meter reading (dB): 98 | 25 | 61

Using the percentage error formula

\text{percentage error} = \frac{|\text{model} - \text{meter}|}{\text{meter}} \times 100\%,

calculate the percentage error for each of the three sound sources, giving your answers to 1 decimal place. [3]

d

Using your results from part (c), advise the engineer whether the rescaled model should be adopted as a reliable measurement tool across different types of sound source, and identify one limitation of using a fixed reference intensity

I_0'

as the basis of the rescaled model. [4]

Solutions

30QuestionConverting between basesAssessment Practice

12 marks~18 minCriterion C

A seismologist models the energy released by earthquakes using the relationship

E = k \cdot 10^{1.5M},

where

M

is the Richter magnitude and

k

is a positive constant. Two earthquakes are recorded: one of magnitude

M_1 = 7.2

and one of magnitude

M_2 = 5.8

.

a

Show that the ratio of energies

\dfrac{E_1}{E_2}

equals

10^{2.1}

, and evaluate this ratio correct to 3 significant figures. [3]

b

A news report claims the magnitude-7.2 earthquake released "about 20 times more energy" than the magnitude-5.8 earthquake. Complete the table below, then determine whether the news report is accurate.

Claim (times more energy): 20
Calculated ratio (3 s.f.): ?
Percentage error (1 d.p.): ?

Use

\text{percentage error} = \frac{|\text{claimed} - \text{calculated}|}{\text{calculated}} \times 100\%.

[4]

c

A second seismologist argues: "Because the Richter scale is logarithmic, every increase of 1 in magnitude multiplies the energy by the same fixed factor." Verify this claim by finding the exact energy multiplication factor for a magnitude increase of 1, and evaluate it correct to 3 significant figures. Advise a science journalist, with reference to your results from parts (a) and (b), whether the Richter scale is suitable for communicating earthquake energy differences to a general public audience. [5]

Solutions

31QuestionApplying change of base formulaAssessment Practice

10 marks~15 minCriterion A

A seismologist compares two historical earthquakes using the formula $\log_{10}(E) = 4.8 + 1.5M$ where $E$ is the energy released in joules and $M$ is the Richter magnitude. The 1960 Valdivia earthquake had magnitude 9.5, and the 2011 Tohoku earthquake had magnitude 9.1.

a

Calculate the energy released by each earthquake, giving your answers in scientific notation with 3 significant figures. [4 marks]

b

Using the change of base formula, find the ratio of the energy of the Valdivia earthquake to that of the Tohoku earthquake. Show your working. [3 marks]

c

The Richter scale is logarithmic. Comment on how this can lead the public to underestimate the relative danger of a slightly higher magnitude earthquake. [3 marks]

Solutions

32QuestionUnderstanding base conversionAssessment Practice

7 marks~11 minCriterion B

The table below shows values of

\log_b(8)

for four different bases

b

.

$b$	2	4	8	16
$\log_b(8)$	3	1.5	1	0.75

a

Describe the pattern in the table, including how

\log_b(8)

changes as

b

increases, and write down the value of

\log_{32}(8)

. [2]

b

The change-of-base formula states

\log_c(x) = \frac{\log_a(x)}{\log_a(c)}, \quad a, c > 0,\ a, c \neq 1,\ x > 0.

Using this formula, find an expression for

\log_b(8)

in terms of

b

, using base 2. Hence verify your answer from part (a) for

b = 32

. [3]

c

A student claims: "Doubling the base always halves the value of

\log_b(8)

."
Using your expression from part (b), justify whether this claim is correct. [2]

Solutions

Logarithms — Free MYP4 Mathematics (Extended) Practice Questions