Laws of Logarithms
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Laws of Logarithms — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionQuotient 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 TT in Figure nn, giving your answer in the form T=an+bT = an + b, where aa and bb are integers. [2]
c
The total number of tiles used to build Figures 1 through nn is given by n2+2nn^2 + 2n. Verify this formula for n=3n = 3, then justify whether this formula confirms that the cumulative total grows faster than the per-figure tile count as nn increases. [3]
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2QuestionQuotient ruleAssessment Practice
10 marks~15 minCriterion A
A company models its monthly profit PP (in thousands of dollars) using the function
P(x)=log⁡2(4x)+log⁡2 ⁣(x2)P(x) = \log_2(4x) + \log_2\!\left(\frac{x}{2}\right)
where xx is the number of units sold (in hundreds) and x>0x > 0.
a
Show that P(x)P(x) can be written in the form P(x)=2log⁡2x+kP(x) = 2\log_2 x + k, where kk is an integer to be found. [3]
b
The table below shows the model's predicted profit alongside observed data at three production levels.

xx (hundreds of units): 1, 2, 4

Predicted PP (thousands of dollars): 1, 3, 5

Actual PP (thousands of dollars): 0.8, 2.6, 5.3

Calculate the percentage error for each value of xx, giving your answers to 2 decimal places, using
percentage error=∣predicted−actual∣actual×100%\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)=5P(x) = 5. Using your result from part (a), solve for the exact value of xx. 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]
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3QuestionQuotient 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 MM to each earthquake, where the intensity II of an earthquake of magnitude MM is given by
I=I0×10M,I = I_0 \times 10^{M},
and I0I_0 is a fixed reference intensity. The magnitude difference between two earthquakes satisfies
ΔM=log⁡10 ⁣(I1I2).\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
percentage error=∣predicted−observed∣observed×100%,\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 I0×104.0I_0 \times 10^{4.0} must trigger an automatic structural inspection. Deduce the minimum magnitude MM 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]
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4QuestionExpanding logarithmic expressionsAssessment Practice
12 marks~18 minCriterion C
A seismologist models earthquake intensity using the Richter scale, where the magnitude MM of an earthquake is defined by
M=log⁡10 ⁣(II0),M = \log_{10}\!\left(\frac{I}{I_0}\right),
where II is the intensity of the earthquake and I0I_0 is a fixed reference intensity.

Two earthquakes are recorded at the same monitoring station. Earthquake A has intensity IA=106 I0I_A = 10^{6}\,I_0 and Earthquake B has intensity IB=109 I0I_B = 10^{9}\,I_0. The energy EE released by an earthquake satisfies E∝I3/2E \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 MB−MA=3M_B - M_A = 3, and hence deduce the ratio IBIA\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 10k10^k where kk 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]
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