Solving Logarithmic Equations

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Solving Logarithmic Equations — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionChecking for extraneous solutionsAssessment Practice

5 marks~8 minCriterion A

The diagram shows part of the graph of

f(x) = \log_2(x + 3)

for

x > -3

Write down the

x

-intercept of the graph of

f

. [1]

Find the value of

x

for which

f(x) = 3

, showing your method clearly. [2]

A second function is defined as

g(x) = \log_2(x + 3) + k

, where

k

is a constant. The

x

-intercept of

g

is at

x = 5

. Justify whether the value of

k

you find is consistent with

g

being a vertical translation of

f

. [2]

Solutions

2QuestionChecking for extraneous solutionsAssessment Practice

6 marks~9 minCriterion D

A chemist monitors the acidity of a river water sample by measuring its pH. The pH of a solution is defined by

\text{pH} = -\log_{10}(c)

where

c

is the hydrogen ion concentration in mol/L. Three samples are recorded.

Sample A — Measured pH:

3

— Predicted

c

(mol/L): ? — Actual

c

(mol/L):

0.00120

Sample B — Measured pH:

5

— Predicted

c

(mol/L): ? — Actual

c

(mol/L):

0.0000085

Sample C — Measured pH:

7

— Predicted

c

(mol/L): ? — Actual

c

(mol/L):

0.000000950

Deduce the predicted hydrogen ion concentration for each sample. Give each answer in standard form to 3 significant figures with units mol/L. [2]

A student claims that a solution with

\text{pH} = -3

is chemically possible. Using

\text{pH} = -\log_{10}(c)

, find the value of

c

that corresponds to

\text{pH} = -3

, and justify whether this value represents a realistic hydrogen ion concentration. [2]

The chemist uses the model to predict concentrations outside the measured range. Assess whether

\text{pH} = -\log_{10}(c)

is a reliable predictive model for river water acidity, supporting your assessment with a specific limitation of the model in this context. [2]

Solutions

3QuestionChecking for extraneous solutionsAssessment Practice

6 marks~9 minCriterion B

The diagram shows three figures in a pattern made from unit squares arranged in an L-shape.

Figure 1 has 2 unit squares, Figure 2 has 6 unit squares, and Figure 3 has 12 unit squares.

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

Find a rule for the number of unit 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 why the rule must contain an

n^2

term by referring to the geometric structure of the figures. [3]

Solutions

4QuestionChecking for extraneous solutionsAssessment Practice

8 marks~12 minCriterion C

An audio engineer is designing a tuning system for a digital synthesiser. The relationship between a frequency ratio

r

and the number of semitones

s

separating two notes is modelled by

\log_2(r) = \frac{s}{12}.

The engineer records observed frequency ratios for three intervals.

$s$ (semitones)	7	12	19
Observed $r$	1.498	2.000	2.973

Calculate the predicted frequency ratio

r

for each value of

s

, giving your answers correct to 3 decimal places. [2]

The percentage error between predicted and observed values is given by

\text{percentage error} = \frac{|\,\text{predicted} - \text{observed}\,|}{\text{observed}} \times 100\%.

Find the percentage error at

s = 7

and at

s = 19

, giving your answers correct to 2 decimal places. Hence determine at which of these two values the model is less accurate. [3]

The engineer claims that because the model predicts exactly

r = 2

when

s = 12

, it is reliable across all values of

s

. Advise the engineer whether this single result provides sufficient justification to use the model with confidence across all values of

s

in the table, referring to your results from parts (a) and (b). [3]

Solutions

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