Exponential–Logarithmic Relationships
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Exponential–Logarithmic Relationships — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionConverting logarithmic equations to exponential formAssessment Practice
5 marks~8 minCriterion C
The graph of f(x)=log⁡2xf(x) = \log_2 x is shown in the diagram for x>0x > 0.
a
Write down the value of f(8)f(8). [1]
b
Find the value of xx for which f(x)=−2f(x) = -2, giving your answer as a fraction. [2]
c
A second function is defined as g(x)=log⁡2(x−3)g(x) = \log_2(x - 3). Justify whether a student who claims the vertical asymptote of g(x)g(x) is x=0x = 0 and the domain is x>0x > 0 is correct. [2]
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Solutions

2QuestionConverting exponential equations to logarithmic formAssessment Practice
6 marks~9 minCriterion D
A sound engineer uses the formula L=10log⁡10 ⁣(II0)L = 10\log_{10}\!\left(\frac{I}{I_0}\right) to model the loudness LL (in decibels, dB) of a sound, where II is the sound intensity and I0I_0 is the reference intensity.

The engineer measures three sounds and records the following data.

Intensity ratio I/I0I/I_0: 100 | 1000 | 50 000

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

Actual LL (dB): 21 | 32 | 49
a
Calculate the three predicted values of LL. Give each answer to the nearest whole number. [2]
b
Find the percentage error for each sound using percentage error=∣ predicted−actual ∣actual×100%.\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]
Question diagram

Solutions

3QuestionConverting 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 SS, the number of squares in Stage nn, where nn 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]
Question diagram

Solutions

4QuestionConverting 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 (nn)1234
Number of squares (SS)371321
a
Write down the number of unit squares in Figure 5. [1]
b
Find a quadratic rule for SS in terms of nn, giving your rule in the form
S=an2+bn+c.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]
Question diagram

Solutions

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