IB · Mathematics: Applications & Interpretation SL & HL · Functions

Log-log & semi-log models: IB Mathematics: Applications & Interpretation questions with answers

Free Log-log & semi-log models questions for IB Mathematics: Applications & Interpretation (SL & HL), HL only, part of Functions. Each question has the answer and the marking points an examiner would look for.

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Log-log & semi-log models practice questions

Question 1Structured · 6 marks

The table shows values of two variables x and y, which are thought to be related by y = axb.

x24681012
y11.253.6126240401586
Table 1.1
(a)

Show that ln y = b ln x + ln a.

[2]
(b)

By finding the equation of the regression line of ln y on ln x, estimate the values of a and b.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)shown
taking logs of both sides M1
ln y = ln a + b ln x AG A1
2
(b)a = 2.45, b = 2.21
transforming the data to ln x and ln y (M1)
ln y = 2.21ln x + 0.895 (A1)
b = 2.21 A1
a = e0.895 = 2.45 A1
4
Question 2Structured · 14 marks

An engineer records the time y seconds for a component to fail under a load of x tonnes.

x24681012
y9.0224.244.262.786.6115
Table 2.1
(a)

Find the equation of the least squares regression line of y on x.

[2]
(b)

Find the sum of the squares of the residuals, SSres, for this linear model.

[2]
(c)

The engineer believes a model of the form y = axb is more appropriate. Using a log-log transformation, find a and b.

[5]
(d)

Find SSres for the model in (c), and hence state which model fits the data better.

[3]
(e)

Use the power model to estimate y when x = 9.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)y = 10.5x − 16.6
GDC (M1)
y = 10.5x − 16.6 A1
2
(b)84.3
Σ(y − ŷ)2 (M1)
84.3 A1
2
(c)a = 3.42, b = 1.41
ln y = ln a + b ln x (M1)
regression of ln y on ln x (M1)
ln y = 1.41ln x + 1.23 A1
b = 1.41 A1
a = 3.42 A1
5
(d)7.66
predicted values from y = axb (M1)
7.66 A1
the power model has the smaller SSres, so fits better R1
3
(e)75.8
3.42 × 91.41 (M1)
75.8 A1
2
Question 3Structured · 4 marks

Data are modelled by y = axn. A graph of ln y against ln x is a straight line with gradient 1.5 and intercept 1.61.

Find a and n, and use the model to predict y when x = 10.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
a = 5, n = 1.5; y = 158
ln y = n ln x + ln a (M1)
n = 1.5 A1
a = 5 A1
158 A1
4
Question 4Structured · 14 marks

An engineer records the time y seconds for a component to fail under a load of x tonnes.

x24681012
y3.384.835.936.597.418.1
Table 4.1
(a)

Find the equation of the least squares regression line of y on x.

[2]
(b)

Find the sum of the squares of the residuals, SSres, for this linear model.

[2]
(c)

The engineer believes a model of the form y = axb is more appropriate. Using a log-log transformation, find a and b.

[5]
(d)

Find SSres for the model in (c), and hence state which model fits the data better.

[3]
(e)

Use the power model to estimate y when x = 7.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)y = 0.457x + 2.84
GDC (M1)
y = 0.457x + 2.84 A1
2
(b)0.346
Σ(y − ŷ)2 (M1)
0.346 A1
2
(c)a = 2.45, b = 0.483
ln y = ln a + b ln x (M1)
regression of ln y on ln x (M1)
ln y = 0.483ln x + 0.894 A1
b = 0.483 A1
a = 2.45 A1
5
(d)0.0274
predicted values from y = axb (M1)
0.0274 A1
the power model has the smaller SSres, so fits better R1
3
(e)6.26
2.45 × 70.483 (M1)
6.26 A1
2
Question 5Structured · 4 marks

Data are modelled by y = axn. A graph of ln y against ln x is a straight line with gradient 0.5 and intercept 1.61.

Find a and n, and use the model to predict y when x = 10.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
a = 5, n = 0.5; y = 15.8
ln y = n ln x + ln a (M1)
n = 0.5 A1
a = 5 A1
15.8 A1
4
Question 6Structured · 4 marks

Data are modelled by y = axn. A graph of ln y against ln x is a straight line with gradient 1.5 and intercept 1.10.

Find a and n, and use the model to predict y when x = 10.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
a = 3, n = 1.5; y = 94.9
ln y = n ln x + ln a (M1)
n = 1.5 A1
a = 3 A1
94.9 A1
4

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in