Introduction to Logarithms
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Introduction to Logarithms — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionEvaluating 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 SS in Figure nn, giving your answer in the form S=an2+bnS = an^2 + bn, where aa and bb are integers to be found. [3]
c
A student claims that Figure nn always contains an odd number of squares. Justify whether this claim is correct for all positive integers nn, referring to the structure of your rule. [3]
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2QuestionEvaluating simple logarithmsAssessment Practice
10 marks~15 minCriterion A
The diagram shows the graph of f(x)=log⁡2xf(x) = \log_2 x for 0<x≤160 < x \leq 16, with three points AA, BB, and CC marked on the curve at x=1x = 1, x=4x = 4, and x=16x = 16 respectively.
a
Find the coordinates of points AA, BB, and CC, giving exact values. [3]
b
A student claims that the point P ⁣(18, k)P\!\left(\tfrac{1}{8},\, k\right) lies on the graph of g(x)=log⁡2x+log⁡2(4x)g(x) = \log_2 x + \log_2(4x). Find the value of kk, showing all steps of your reasoning. [3]
c
The equation log⁡2(x+3)+log⁡2(x−1)=5\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]
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3QuestionEvaluating simple logarithmsAssessment Practice
12 marks~18 minCriterion C
Two major earthquakes are compared using the Richter scale formula
M=log⁡10 ⁣(II0),M = \log_{10}\!\left(\frac{I}{I_0}\right),
where MM is the magnitude, II is the intensity of the earthquake, and I0I_0 is a fixed reference intensity.

The 1906 San Francisco earthquake had magnitude M=7.8M = 7.8, the 2011 Tohoku earthquake had magnitude M=9.1M = 9.1, and the 1960 Valdivia earthquake had magnitude M=8.6M = 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,\log_{10}(E) = 4.8 + 1.5M,
where EE 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=7.8M = 7.8) for three earthquakes.

Earthquake — Magnitude — Energy ratio (relative to San Francisco)
San Francisco7.81.00
Valdivia8.6RR
Tohoku9.1SS


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]
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4QuestionNatural 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 tt years under Option A is modelled by
A(t)=5000(1+0.04812)12t,A(t) = 5000\left(1 + \frac{0.048}{12}\right)^{12t},
and under Option B by
B(t)=5000(1+0.04754)4t.B(t) = 5000\left(1 + \frac{0.0475}{4}\right)^{4t}.

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

Year (tt)51020
Actual balance (dollars)6283789112530
a
Calculate the balance predicted by Option A at t=5t = 5, t=10t = 10, and t=20t = 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=5t = 5, t=10t = 10, and t=20t = 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
percentage error=∣predicted−actual∣actual×100%.\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=5t = 5, t=10t = 10, and t=20t = 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]
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