Logarithmic & exponential functions: A Level Mathematics questions with answers
Free Logarithmic & exponential functions questions for A Level Mathematics (9709), part of Pure Mathematics 3. Each question has the answer and the marking points an examiner would look for.
Logarithmic & exponential functions practice questions
Question 1Structured · 11 marksA* challenge
The equation 22x + 1 − k(2x) + 8 = 0, where k is a real constant, has roots x₁ and x₂.
(a)
Find the set of values of k for which the equation has two distinct real roots for x.
[4]
(b)
Find the value of k for which x₁2 + x₂2 = 4.
[5]
(c)
Find the set of possible values of x₁2 + x₂2 as k varies over the set found in (a).
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
k > 8
u = 2x: 2u2 − ku + 8 = 0 B1 both roots in u must be positive (since 2x > 0) M1 k2 > 64 A1 k > 8 only (k < −8 gives negative u) A1
4
(b)
k = 10 (x = 0 and 2)
product of roots in u is 4 M1 2x₁ + x₂ = 4, so x₁ + x₂ = 2 A1 (x₁ + x₂)2 − 2x₁x₂ = 4 gives x₁x₂ = 0 M1 x = 0 and 2, so u = 1 and 4 A1 k = 2(1 + 4) = 10 A1
5
(c)
x₁2 + x₂2 > 2
x₁2 + x₂2 = 2 + ½(x₁ − x₂)2 (or 4 − 2x₁x₂ with x₁x₂ < 1) M1 all values greater than 2 (equality only when the roots coincide, k = 8) A1
2
Question 2Structured · 5 marksA* challenge
The variables x and y satisfy the equation y = A bx, where A and b are constants. The graph of ln y against x is a straight line passing through the points (1.1, 2.2) and (5, 4.36).
Find the values of A and b, giving your answers correct to 3 significant figures.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
A = 4.91, b = 1.74
ln y = ln A + x ln b stated B1 gradient from the two points M1 ln b = 0.554 A1 intercept found and exponentiated M1 A = 4.91, b = 1.74 A1
5
Question 3Structured · 3 marks
Solve the equation 3x+1 = 5x, giving your answer correct to 3 significant figures.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
x = 2.15
Take logs of both sides M1 (x + 1) ln 3 = x ln 5 M1 2.15 A1
u = 5xM1 u2 − 126u + 125 = 0 A1 u = 1, u = 125 A1 x = 0, x = 3 A1
4
Question 5Structured · 5 marksA* challenge
The variables x and y satisfy the equation y = A xn, where A and n are constants. The graph of ln y against ln x is a straight line passing through the points (1.3, 2.01) and (3.8, 4.39).
Find the values of A and n, giving your answers correct to 3 significant figures.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
A = 2.16, n = 0.952
ln y = ln A + n ln x stated B1 gradient from the two points M1 n = 0.952 A1 intercept found and exponentiated M1 A = 2.16, n = 0.952 A1
5
Question 6Structured · 3 marks
Solve the equation 2x+1 = 5x, giving your answer correct to 3 significant figures.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
x = 0.756
Take logs of both sides M1 (x + 1) ln 2 = x ln 5 M1 0.756 A1
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Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in