Laws of Logarithms

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Laws of Logarithms — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionCombining multiple log lawsAssessment Practice

5 marks~8 minCriterion B

The diagram shows three figures in 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.

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

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. [2]

Verify your rule for Figure 3. Justify whether the

n^2

term is necessary by referring to the structure of the figures. [2]

Solutions

2QuestionCondensing logarithmic expressionsAssessment Practice

6 marks~9 minCriterion C

The diagram shows part of the graph of

f(x) = \log_2 x

for

x > 0

Write down the value of

f(8)

. [1]

Given that

\log_2 3 + \log_2 5 - \log_2 k = f(1)

, find the value of

k

. [2]

A student claims that

f(x) = 2

has the same solution as

f(x^2 - 3x) = f(4)

. Justify whether the student is correct. [3]

Solutions

3QuestionCombining multiple log lawsAssessment Practice

7 marks~11 minCriterion A

Let

f(x) = \log_2(8x^3)

, where

x > 0

Show that

f(x) = 3 + 3\log_2 x

. [2]

Find the value of

x

for which

f(x) = 9

. [2]

The graphs of

y = f(x)

and

y = -3\log_2 x + 9

intersect at exactly one point. Find the coordinates of that point. [3]

Solutions

4QuestionCombining multiple log lawsAssessment Practice

8 marks~12 minCriterion D

A sound engineer is designing an outdoor concert stage and needs to predict combined sound levels from multiple speaker stacks. The combined sound intensity level, in decibels (dB), from two independent sources is modelled by

L_{\text{total}} = 10\log_{10}\!\left(10^{L_1/10} + 10^{L_2/10}\right)

where

L_1

and

L_2

are the individual sound levels in dB. The main speaker stack measures

L_1 = 85

dB and the monitor stack measures

L_2 = 92

dB.

Predicted and actual combined levels at three stage positions are given below.

Position	A	B	C
Predicted $L_{\text{total}}$ (dB)	92.8	89.3	94.1
Actual measured (dB)	91.2	90.8	95.6

Calculate the percentage error between the predicted and actual combined level at position A, where the predicted value is 92.8 dB and the actual measured value is 91.2 dB. Give your answer correct to 2 decimal places.

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

[2]

The engineer proposes adding a third speaker stack at 85 dB to the existing combination of 85 dB and 92 dB. Using the formula, find the new combined level of all three sources. Give your answer correct to 1 decimal place. [3]

The percentage errors at positions A, B and C are 1.75\%, 1.65\% and 1.00\% respectively. Advise the engineer whether this model is reliable enough to use for stage safety planning, and identify one limitation of using this logarithmic model to represent human perception of loudness. [3]

Solutions

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