Solving Logarithmic Equations

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Solving Logarithmic Equations — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionSolving 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

Write down

A_4

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

Find a rule for

A_n

in the form

A_n = an + b

, where

a

and

b

are integers. [3]

Justify why

A_n

must be a linear function of

n

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

Solutions

2QuestionSolving 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

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]

Hence find the value of

x

that satisfies

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

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

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

3QuestionChecking 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)	$3.2 \times 10^{9}$	$3.5 \times 10^{12}$	$8.0 \times 10^{17}$

Calculate the predicted energy

E

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

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]

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]

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

4QuestionSolving 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

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]

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]

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

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