Exponential and Logarithmic Functions · CAPE Pure Mathematics Unit 1

57 past-paper questions on Exponential and Logarithmic Functions, part of Basic Algebra and Functions, from every CAPE Pure Mathematics Unit 1 paper on Quelpr.

1. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The exact value of \left(\frac{25}{16}\right)^{-\frac{1}{2}} is
2. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 13\log_2 2q - 2\log_2 3q + 1 expressed as a single logarithm in its SIMPLEST form is
3. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1The annual growth, g(x), (in thousands) of the population in a country for x years is represented by g(x) = 2^x. In how many years will a growth of 32 thousand be achieved?
4. 1(d)6 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Solve the logarithmic equation \log_5(x + 2) + \log_5(x + 6) = 1.
5. 2(b)8 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Solve the equation 6 - \frac{7}{2^{2x}} - \frac{3}{4^{2x}} = 0.
6. 3(b)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Without the use of calculators or tables, show that 4^2 / (sqrt(2) * 8^(-1/3)) = 2^4 (sqrt(2)).
7. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1\log_5 5\sqrt{5} is equal to
8. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1If \log_a 4 + \log_a x - \log_a 7 = 2, then the value of x is
9. Q31 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 13^{\log_3 5} =
10. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1The value of \log_{\sqrt{6}} 36 is
11. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The exact value of \left(\frac{25}{16}\right)^{-\frac{1}{2}} is
12. Q111 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The value of \log_{\sqrt{6}} 36 is
13. Q141 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The annual growth, g(x), (in thousands) of the population over x years is represented by g(x) = 2^x. Over how many years will an annual growth of 32 thousand be achieved?
14. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1If \log_a 4 + \log_a x - \log_a 7 = 2, then the value of x is
15. Q111 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1The annual growth, g(x), (in thousands) of the population over x years is represented by g(x) = 2^x. Over how many years will an annual growth of 32 thousand be achieved?
16. Q121 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1\log 15 - \log 6 + \frac{1}{2}\log\frac{4}{25} =
17. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1\ln x^y =
18. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1If \log_a 4 + \log_a x - \log_a 7 = 2, then the value of x is
19. Q131 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 13^{\log_3 5} =
20. Q51 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 13\log_2 2q - 2\log_2 3q + 1 expressed as a single logarithm in its SIMPLEST form is
21. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1The annual growth, g(x), (in thousands) of the population in a country for x years is represented by g(x) = 2^x. In how many years will a growth of 32 thousand be achieved?
22. Q121 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1\log 15 - \log 6 + \frac{1}{2} \log \frac{4}{25} =
23. Q141 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 13^{\log_3 5} =
24. 1(d)(i)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Solve the logarithmic equation \log_3(x^2 - 9) - \log_3(x + 3) = 3.
25. 1(c)8 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Solve the logarithmic equation log_5(x) - 4 log_x(5) = -3.
26. 2(a)(i)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Complete the table of values for h(x) = 5 - 2^(x+1) for x = 0, 1, 2, 3, 4.
27. 2(a)(ii)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2On the grid provided on page 9, plot the graph of h(x) = 5 - 2^(x+1).
28. 2(d)(i)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Solve for x: 3^(4x + 1) = 177 147.
29. 2(d)(ii)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Solve for x: log₂(x + 6) + log₂(x + 2) = 5.
30. 1(b)8 marks· Pure Mathematics · Unit 1 Q1 1(b)Solve, for x and y, the simultaneous equations log (x-1) + 2 log y = 2 log 3 and log x + log y = log 6.
31. 1(c)8 marks· Pure Mathematics · Unit 1 Q1 1(c)Solve the equation log₂ x = 1 + log₂ 2x, x > 0.
32. 1(c)(i)4 marks· Pure Mathematics · Unit 1 Q1 1(c)(i)If 2^(x²) = 16^(x-1), find x.
33. 1(c)(i)6 marks· Pure Mathematics · Unit 1 Q1 1(c)(i)By substituting y = log₂x, or otherwise, solve, for x, the equation √log₂x = log₂(√x).
34. 1(d)7 marks· Pure Mathematics · Unit 1 Q1 1(d)By using y = 2ˣ, or otherwise, solve 4ˣ - 3 (2ˣ⁺¹) + 8 = 0.
35. 1(d)6 marks· Pure Mathematics · Unit 1 Q1 1(d)Solve the logarithmic equation log₄(2x + 2) - log₄(x + 1) = 1.
36. 1(d)6 marks· Pure Mathematics · Unit 1 Q1 1(d)Solve the logarithmic equation log₂ x + log₄ x + log₁₆ x = 7.
37. 1(e)(i)4 marks· Pure Mathematics · Unit 1 Q1 1(e)(i)Using a scale of 2 cm to represent 10 minutes on the x-axis and 2 cm to represent 20 bacteria on the y-axis, draw the growth curve for the treatment stage.
38. 1(e)(ii)1 mark· Pure Mathematics · Unit 1 Q1 1(e)(ii)Hence, estimate the number of bacteria present after the first 35 minutes of treatment.
39. 2(a)6 marks· Pure Mathematics · Unit 1 Q2 2(a)Solve the following equation for x: log₂(10 - x) + log₂x = 4.
40. 2(a)(i)5 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)Given that a² + b² = 14ab, prove that ln((a+b)/4) = (1/2)(ln a + ln b).
41. 2(a)(i)5 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)Show that 1/logₐx may be rewritten as ln a / ln x, where a > 0, a ≠ 1.
42. 2(a)(ii)4 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Hence, or otherwise, solve the equation: 1/log₂x + 1/log₃x + 1/log₄x + 1/log₅x = 1.
43. 2(a)(ii)6 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Solve the equation 2ˣ + 3(2⁻ˣ) = 4. [Your response may be expressed in terms of logarithms.]
44. 2(b)5 marks· Pure Mathematics · Unit 1 Q2 2(b)Given that a² + b³ + 3a²b = 5ab², show that 3 log((a+b)/2) = log a + 2 log b.
45. 2(b)(i)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)Prove that logₙm = log₁₀m / log₁₀n, for m, n ∈ N.
46. 2(b)(i)7 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)Solve 3 - 4/(9)ˣ - 4/(81)ˣ = 0.
47. 2(b)(i)5 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)Solve for x the equation x^(1/3) - 4x^(-1/3) = 3.
48. 2(b)(ii)6 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)Hence, given that y = (log₂ 3) (log₃ 4) (log₄ 5) ... (log₃₁ 32), calculate the exact value of y.
49. 2(b)(ii)5 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)Find x such that log₂(x + 3) + log₂(x - 1) = 1.
50. 2(b)(iii)3 marks· Pure Mathematics · Unit 1 Q2 2(b)(iii)Without the use of calculators or tables, evaluate log₁₀(1/2) + log₁₀(2/3) + log₁₀(3/4) + ... + log₁₀(8/9) + log₁₀(9/10).
51. 2(c)1 mark· Pure Mathematics · Unit 1 Q2 2(c)Without the use of calculators or tables, evaluate
52. 2(c)(i)4 marks· Pure Mathematics · Unit 1 Q2 2(c)(i)Solve EACH of the following equations: eˣ + 1/eˣ - 2 = 0
53. 2(c)(i)1 mark· Pure Mathematics · Unit 1 Q2 2(c)(i)Determine the number of bacteria present at t = 0.
54. 2(c)(i)3 marks· Pure Mathematics · Unit 1 Q2 2(c)(i)log₁₀(1/3) + log₁₀(3/5) + log₁₀(5/7) + log₁₀(7/9) + log₁₀(9/10)
55. 2(c)(ii)4 marks· Pure Mathematics · Unit 1 Q2 2(c)(ii)Determine the time required to triple the number of bacteria.
56. 2(c)(ii)4 marks· Pure Mathematics · Unit 1 Q2 2(c)(ii)Solve EACH of the following equations: log₂(x + 1) – log₂(3x + 1) = 2
57. 5(d)(ii)3 marks· Pure Mathematics · Unit 1 Q5 5(d)(ii)Determine the time required for the bacteria population to double in size.