TMUA Exponentials and Logarithms — Practice Questions by Topic

Worked example

This question is also in the list below, with the solution folded. It is opened here so the method is on the page before the other questions.

Solve log₂(x + 3) + log₂(x − 1) = 5.

1. A. −7
2. B. 2
3. C. 5
4. D. 15
5. E. 29

Answer: C. 5

Worked solution. The logarithms are defined for x > 1. Their sum is log₂((x + 3)(x − 1)) = 5, so (x + 3)(x − 1) = 32. Then x² + 2x − 35 = 0, or (x + 7)(x − 5) = 0. The roots are x = −7 and x = 5, and only x = 5 satisfies x > 1. Check: log₂(8) + log₂(4) = 3 + 2 = 5. The solution is 5.

Why the other options look right. A keeps x = −7, which makes x − 1 negative. B sets (x + 3)(x − 1) equal to 5 instead of 2⁵, and the in-domain root is 2. D adds the arguments and sets (x + 3) + (x − 1) = 32, so x = 15. E sets only x + 3 equal to 32, so x = 29.

Paper 1 questions

Paper 1 is Applications of Mathematical Knowledge. Calculators are not allowed.

1. Q1. Solve log₂(x + 3) + log₂(x − 1) = 5.

   Free · worked solution included

   A−7B2C5D15E29

   Answer and worked solution

   Answer: C. 5

   Worked solution. The logarithms are defined for x > 1. Their sum is log₂((x + 3)(x − 1)) = 5, so (x + 3)(x − 1) = 32. Then x² + 2x − 35 = 0, or (x + 7)(x − 5) = 0. The roots are x = −7 and x = 5, and only x = 5 satisfies x > 1. Check: log₂(8) + log₂(4) = 3 + 2 = 5. The solution is 5.

   Why the other options look right. A keeps x = −7, which makes x − 1 negative. B sets (x + 3)(x − 1) equal to 5 instead of 2⁵, and the in-domain root is 2. D adds the arguments and sets (x + 3) + (x − 1) = 32, so x = 15. E sets only x + 3 equal to 32, so x = 29.

2. Q2. Solve 8^{x + 1} = 4^{2x − 1}.

   Free · worked solution included

   A−4B1C2D3E5

   Answer and worked solution

   Answer: E. 5

   Worked solution. Write 8 = 2³ and 4 = 2². The equation becomes 2^{3(x + 1)} = 2^{2(2x − 1)}, so 3x + 3 = 4x − 2. Therefore x = 5.

   Why the other options look right. A treats 4 as 2¹, so 3x + 3 = 2x − 1 and x = −4. B flips the sign of the constant on the right and solves 3x + 3 = 4x + 2, so x = 1. C equates the original exponents, x + 1 = 2x − 1, so x = 2. D writes the left exponent as 3x + 1 instead of 3(x + 1), so 3x + 1 = 4x − 2 and x = 3.

3. Q3. A colony of algae triples every 5 hours. It starts with 200 cells. How many cells are there after 15 hours?

   Free · worked solution included

   A600B1800C3000D5400E16200

   Answer and worked solution

   Answer: D. 5400

   Worked solution. Fifteen hours is 15/5 = 3 tripling periods. The population is 200 × 3³ = 200 × 27 = 5400.

   Why the other options look right. A multiplies by one factor of 3, giving 600. B uses two triplings, 200 × 9 = 1800. C multiplies the starting population by the 15 hours, giving 3000. E uses four triplings, 200 × 81 = 16200.

4. Q4. Find the real solution of log₂(x + 2) + log₂(x − 4) = 4.

   Free · worked solution included

   A−4B6C9D14E20

   Answer and worked solution

   Answer: B. 6

   Worked solution. The logarithms require x + 2 > 0 and x − 4 > 0, so x > 4. Adding the logs gives log₂((x + 2)(x − 4)) = 4, hence (x + 2)(x − 4) = 16. Expanding produces x² − 2x − 8 = 16, so x² − 2x − 24 = 0, which factors as (x − 6)(x + 4) = 0. The root x = −4 makes an argument negative, and x = 6 lies in the domain. The solution is 6.

   Why the other options look right. A is the other root of the quadratic, but log₂(−4 + 2) is undefined. C adds the arguments instead of multiplying them: (x + 2) + (x − 4) = 16 gives x = 9. D sets only the first argument equal to 2⁴, so x + 2 = 16 and x = 14. E sets only the second argument equal to 16, so x − 4 = 16 and x = 20.

5. Q5. Solve 9^x × 27^{x − 1} = 3^{x + 7}.

   Free · worked solution included

   A5/2B1C4/3D2E3

   Answer and worked solution

   Answer: A. 5/2

   Worked solution. Write every power with base 3. Then 9^x = (3²)^x = 3^2x and 27^{x − 1} = (3³)^{x − 1} = 3^{3x − 3}, so the left side is 3^2x × 3^{3x − 3} = 3^{5x − 3}. Equating exponents with the right side gives 5x − 3 = x + 7, so 4x = 10 and x = 5/2. Check: both sides equal 3^19/2 when x = 5/2.

   Why the other options look right. B reaches 5x − 3 = x + 7 but moves the 3 across without changing its sign, so 4x = 7 − 3 = 4 and x = 1. C multiplies the bases and adds the exponents, writing 9^x × 27^{x − 1} as 243^{2x − 1} = 3^{10x − 5}, so 10x − 5 = x + 7 and x = 4/3. D writes (3³)^{x − 1} as 3^{3x − 1}, multiplying only the x by 3, so 2x + 3x − 1 = x + 7 and x = 2. E adds the powers instead of multiplying them when changing the base, writing 9^x as 3^{x + 2} and 27^{x − 1} as 3^{(x − 1) + 3} = 3^{x + 2}, so 2x + 4 = x + 7 and x = 3.

6. Q6. Solve 2^{x + 1} + 2^{x − 1} = 40.

   Free · worked solution included

   A3B4C5D8E20

   Answer and worked solution

   Answer: B. 4

   Worked solution. Factor out the smaller power: 2^{x + 1} + 2^{x − 1} = 2^{x − 1}(4 + 1) = 5 × 2^{x − 1}. This equals 40, so 2^{x − 1} = 8 = 2³. Hence x − 1 = 3 and x = 4.

   Why the other options look right. A stops at the exponent x − 1 = 3 and does not add 1. C factors out 2^{x + 1} instead: 2^{x + 1}(1 + 1/4) = 40 gives 2^{x + 1} = 32, and then it reports the exponent x + 1 = 5 instead of x. D reports 8, the value of 2^{x − 1}, rather than x. E replaces each power of 2 by its exponent and solves (x + 1) + (x − 1) = 40, so 2x = 40 and x = 20.

7. Q7. Solve log₂(x² − 1) − log₂(x − 1) = 3.

   Free · worked solution included

   A8B1C9D3E7

   Answer and worked solution

   Answer: E. 7

   Worked solution. Both logarithms are defined only for x > 1. For those x, the left side is log₂((x² − 1)/(x − 1)) = log₂(x + 1). So x + 1 = 2³ = 8 and x = 7. This is greater than 1, and (49 − 1)/(7 − 1) = 8, which checks. The solution is 7.

   Why the other options look right. A reports 2³, the value of x + 1, rather than x. B clears the logarithms to (x − 1)(x + 1) = 8(x − 1) and reports the root x = 1 without checking that it makes log₂(x − 1) undefined. C combines the logs correctly but simplifies (x² − 1)/(x − 1) to x − 1 instead of x + 1, so x − 1 = 8 and x = 9. D solves only log₂(x² − 1) = 3, so x² = 9, and reports the positive square root x = 3.

8. Q8. Solve log₃(x − 2) + log₃(2x + 3) = 2.

   Free · worked solution included

   A6B3C−5/2D8/3E5/2

   Answer and worked solution

   Answer: B. 3

   Worked solution. The logarithms require x − 2 > 0 and 2x + 3 > 0, so x > 2. Their sum is log₃((x − 2)(2x + 3)) = 2, and the product equals 3² = 9. Expanding gives 2x² − x − 15 = 0, so x = (1 ± 11)/4. The roots are x = 3 and x = −5/2. Only x = 3 is greater than 2, and (3 − 2)(6 + 3) = 9, which checks. The solution is 3.

   Why the other options look right. A uses the quadratic formula with denominator 2 instead of 4 and keeps (1 + 11)/2 = 6. C solves 2x² − x − 15 = 0 correctly but keeps the root x = −5/2 without checking that it makes x − 2 negative. D combines the logarithms as log₃((x − 2) + (2x + 3)) = 2, adding the arguments instead of multiplying them, so 3x + 1 = 9 and x = 8/3. E expands (x − 2)(2x + 3) as 2x² + x − 6 with the wrong sign on the x term, so 2x² + x − 15 = 0 gives x = 5/2 or x = −3, and keeps x = 5/2 since it exceeds 2.

9. Q9. Solve log₂(x + 5) − log₂(x − 2) > 2.

   Free · correct letter only

   Ax > 2B2 < x < 13/3Cx < 13/3Dx > 13/3E2 < x < 9

   Answer

   Answer: B. 2 < x < 13/3

   The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

10. Q10. The equation 3^x + 3^−x = 5/2 has exactly one positive solution. What is it?

    Free · correct letter only

    Alog₃(5/2)B2Clog₃(1/2)Dlog₃(2)E1/2

    Answer

    Answer: D. log₃(2)

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

11. Q11. Solve log₂(x² − 2x − 3) < log₂(2x + 6).

    Free · correct letter only

    A2 − √13 < x < −1B3 < x < 2 + √13Cx > 3D−3 < x < −1 or x > 3E2 − √13 < x < −1 or 3 < x < 2 + √13

    Answer

    Answer: E. 2 − √13 < x < −1 or 3 < x < 2 + √13

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

12. Q12. Find the solution of log₃(x + 6) − log₃(x − 2) = 2.

    Free · correct letter only

    A2B3C6D9E10

    Answer

    Answer: B. 3

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

13. Q13. Find the complete set of real x such that log₂(x + 2) + log₂(x − 2) < 5.

    Free · correct letter only

    Ax > 6B−6 < x < 6Cx > 2D2 < x < 6E0 < x < 6

    Answer

    Answer: D. 2 < x < 6

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

14. Q14. The three equations log₂(log₅ x) = 0, log₅(log₃ y) = 0 and log₃(log₂ z) = 0 all hold, with x, y and z positive. What is x + y + z?

    Free · correct letter only

    A3B9C10D30E276

    Answer

    Answer: C. 10

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

15. Q15. The equation 9^2x − 5 × 3^2x+1 + 36 = 0 has two real solutions. Find the positive difference of those solutions.

    Free · correct letter only

    A9Blog₁₀(4) / log₁₀(9)C1Dlog₁₀(4) / log₁₀(3)Elog₁₀(4)

    Answer

    Answer: B. log₁₀(4) / log₁₀(9)

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

16. Q16. The numbers x and y satisfy x > 0, y > 1, log₁₀(y − 1) = 2 log₁₀(x) − log₁₀(2) and log₁₀(y − 5x + 4) = 0. What are the possible values of y?

    Free · correct letter only

    A21 ± 5√15B5 ± √17C7 and 37D22 ± 5√17E22 ± √17

    Answer

    Answer: D. 22 ± 5√17

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

17. Q17. The solution of the simultaneous equations 3^x + 2 times 3^y = 4 and 3^2x − 4 times 3^2y = 8 is x = p and y = q. Find the value of p − q.

    Free · correct letter only

    A6B5/2Clog base 3 of (3/2)Dlog base 3 of (5/2)Elog base 3 of 6

    Answer

    Answer: E. log base 3 of 6

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

18. Q18. Positive numbers x and y, neither equal to 1, satisfy log₃(xy) = 6 and log_x(y) + log_y(x) = 5/2. What is the value of x + y?

    Free · correct letter only

    A90B6C18D42E729

    Answer

    Answer: A. 90

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

19. Q19. Find the sum of the real solutions of 5^x − (√5)^{x + 2} + 6 = 0.

    Free · correct letter only

    A2B5Clog₅(6)D2 log₅(6)E(1/2) log₅(6)

    Answer

    Answer: D. 2 log₅(6)

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

20. Q20. Find the sum of the real values of x that satisfy both log₂(x y²) = 1 and (log₂ x)(log₂ y) = −6.

    Free · correct letter only

    A1B16C129/16D129/8E2

    Answer

    Answer: D. 129/8

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

21. Q21. Find the maximum value of 9^{sin x} − 6 × 3^{sin x} + 4 for real x.

    Free · correct letter only

    A−5B19/9C100/9D4EThere is no maximum value

    Answer

    Answer: B. 19/9

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

22. Q22. Find the real solution of 9^2x / 3^4x = 1/3.

    Free · correct letter only

    Alog₂(1 + √2)Blog₂((1 + √5)/2)C1D0Elog_{1 + √2} 2

    Answer

    Answer: A. log₂(1 + √2)

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

23. Q23. What is the maximum value of 1/(3^2x − 6(3^x) + 13)?

    Free · correct letter only

    A1/13B1/8C4D1E1/4

    Answer

    Answer: E. 1/4

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

24. Q24. Given that 3^2x = 9^{y + 2} and 27^x = 3^{y + 12}, what is the value of x + y?

    Free · correct letter only

    A5B8C28D3E2

    Answer

    Answer: B. 8

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

25. Q25. Find the positive difference between the two real values of x for which (log₃ x)⁴ + 6 (log₃ (1/x))² − 7 = 0.

    Free · correct letter only

    A2B3C10/3D8/3E1

    Answer

    Answer: D. 8/3

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

26. Q26. What is the minimum value of 9^x − 6 × 3^x + 5?

    Free · correct letter only

    A−9B−4C5D14E0

    Answer

    Answer: B. −4

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

27. Q27. The integral of x from log base 3 of 4 to log base 3 of 36 equals log base 3 of M. What is M?

    Free · correct letter only

    A9B12C36D144E20736

    Answer

    Answer: D. 144

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

28. Q28. Evaluate the sum from n = 1 to n = 40 of log base 2 of (5^{3 − n}).

    Free · correct letter only

    A−820 log₂ 5B−700 log₂ 5C700 log₂ 5D−660 log₂ 5E120 log₂ 5

    Answer

    Answer: B. −700 log₂ 5

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

29. Q29. P(x) = 2ˣ + 4 and Q(x) = (1/4)(2ˣ)² − 3 × 2ˣ + 10. Find the largest x such that P(x) and Q(x) are in the ratio 4 : 1.

    Free · correct letter only

    Alog base 2 of (8 + 2√10)B9C2Dlog base 2 of 13Elog base 2 of 9

    Answer

    Answer: E. log base 2 of 9

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

30. Q30. The difference between the maximum and minimum values of f(x) = a^{cos x}, where a > 0, is 5. Find the sum of the possible values of a.

    Free · correct letter only

    A0B√29C5D(5 + √29)/2E√21

    Answer

    Answer: B. √29

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

31. Q31. Given that y = 2 log₁₀(3 − x) for x < 3, which expression equals x?

    Free · correct letter only

    A3 − log₁₀(y/2)B3 + 10^y/2C3 − 10^y/2D10^y/2 − 3E3 − 10^2y

    Answer

    Answer: C. 3 − 10^y/2

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

32. Q32. The numbers p and q are positive and different from 1, with p ≠ q², and p^{x + 1} = q^2x. Which expression equals x?

    Free · correct letter only

    Alog₁₀(p) / (log₁₀(q) − log₁₀(p))Blog₁₀(p) / (log₁₀(p) − 2 log₁₀(q))C0Dlog₁₀(p) / (2 log₁₀(q) + log₁₀(p))Elog₁₀(p) / (2 log₁₀(q) − log₁₀(p))

    Answer

    Answer: E. log₁₀(p) / (2 log₁₀(q) − log₁₀(p))

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

33. Q33. What is the sum of the real roots of 3^2x − 10 × 3^x + 21 = 0?

    Free · correct letter only

    A10Blog₃(10)Clog₃(7/3)Dlog₃(21)Elog₉(21)

    Answer

    Answer: D. log₃(21)

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

34. Q34. For positive x, the graph of log₁₀(y) against log₁₀(x) is the straight line log₁₀(y) = 2 log₁₀(x) + log₁₀(3). Which equation relates y and x?

    Free · correct letter only

    Ay = 2x + 3By = 3 × 2^xCy = 3x²Dy = x² + 3Ey = 6x

    Answer

    Answer: C. y = 3x²

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

35. Q35. Solve log₃(x) + log₃(x − 8) = 2.

    Free · correct letter only

    A−1B17/2C9D4 + 2√6E4 + √22

    Answer

    Answer: C. 9

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

36. Q36. Solve log₂(x) = 5.

    Free · worked solution included

    A10B16C25D32E64

    Answer and worked solution

    Answer: D. 32

    Worked solution. log₂(x) = 5 means 2⁵ = x. So x = 32.

    Why the other options look right. A multiplies the base by the log result, 2 × 5. B multiplies only four factors of 2, 2 × 2 × 2 × 2 = 16, losing a factor when counting to five. C squares the result, 5². E multiplies six factors of 2, 2 × 2 × 2 × 2 × 2 × 2 = 64, counting one factor too many.

37. Q37. The curve y = ax² + 1 passes through the points (1, log₂ p) and (log₂ p + 1, 33), where p is positive. Find p.

    Free · correct letter only

    A2B3C4D8E16

    Answer

    Answer: D. 8

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

38. Q38. Which of the following five numbers is the greatest? Here 3√3 means 3 times √3.

    Free · correct letter only

    Alog₂ 48B√30C3√3Dlog₃ 180E(9 + √5)/3

    Answer

    Answer: A. log₂ 48

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

39. Q39. Let p, q and r be positive real numbers with p ≠ 1 and q ≠ 1. The equation log_p((p^x²)³) + (log_q r)x + (log₂ 8)x + log_q r = 0 has a repeated root when

    Free · correct letter only

    Ar = 3qBq³ r = 1Cq⁹ r = 1Dr = q³Eq = r³

    Answer

    Answer: D. r = q³

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

40. Q40. Find the nonzero real solution of the equation in which 3 raised to the power 4^x, divided by 729 raised to the power 2^x, equals 1/243.

    Free · correct letter only

    A5Blog base 2 of 5Clog base 5 of 2D0Elog base 2 of 6

    Answer

    Answer: B. log base 2 of 5

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

41. Q41. The numbers x, y and z are greater than 1, and b > 1. The logarithm base b of (x y³) equals 5, the logarithm base b of (x² y z) equals 8, and the logarithm base b of (x y z²) equals 9. Which statement is true?

    Free · correct letter only

    Ab = 1Bb = xCb = yDThere is not enough information to determine bEb = z

    Answer

    Answer: C. b = y

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

42. Q42. How many distinct real solutions does the following equation have? The logarithm to the base x² − 5 of (x⁴ − x³ − 8x² + 3x + 25) equals 2.

    Free · correct letter only

    A0B1C2D3E4

    Answer

    Answer: B. 1

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

43. Q43. Evaluate 27^2/3 + 32^−4/5.

    Free · correct letter only

    A97/16B145/16C25D−7E9/16

    Answer

    Answer: B. 145/16

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

44. Q44. If log_a b = 3 and log_b c = 4, what is log_a c?

    Free · correct letter only

    A7B64C12D1/12E3/4

    Answer

    Answer: C. 12

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

45. Q45. Let p = log₂ 3. Which expression is equal to log₁₂ 18?

    Free · correct letter only

    A(2 + p)/(1 + 2p)B(1 + 2p)/(2 + p)Cp − 1D3/2E(1 + p²)/(2 + p)

    Answer

    Answer: B. (1 + 2p)/(2 + p)

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

46. Q46. Work out the exact value of 8 to the power 2/3, plus 81 to the power −1/2.

    Free · correct letter only

    A37/9B4C13/3D35/9E13

    Answer

    Answer: A. 37/9

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

47. Q47. You are told that log base 9 of 3 equals 1/2 and log base 3 of 243 equals 5. Find log base 9 of 243.

    Free · correct letter only

    A10B5/2C11/2D5E5/4

    Answer

    Answer: B. 5/2

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

48. Q48. Given that log₃ 9 = 2 and log₉ 729 = 3, find log₃ 729.

    Free · correct letter only

    A5B8C9D18E6

    Answer

    Answer: E. 6

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

49. Q49. Work out 81^3/4 + 8^−2/3.

    Free · correct letter only

    A109/4B31C23D13/4E27/4

    Answer

    Answer: A. 109/4

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

50. Q50. You are told that log₂ 8 = 3 and log₈ 32 = 5/3. Work out log₂ 32.

    Free · correct letter only

    A14/3B5/9C8D5E3

    Answer

    Answer: D. 5

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

51. Q51. Evaluate 16^3/4 + 27^−2/3.

    Free · correct letter only

    A17B71/9C73/9D25/3E37/9

    Answer

    Answer: C. 73/9

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

52. Q52. You are told that the logarithm of 32 in base 4 is 5/2, and that the logarithm of 1024 in base 32 is 2. Find the logarithm of 1024 in base 4.

    Free · correct letter only

    A9/2B5C4/5D10E25/4

    Answer

    Answer: B. 5

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

53. Q53. Evaluate 27^2/3 + 81^−1/4.

    Free · correct letter only

    A28/3B6C10/3D12E55/3

    Answer

    Answer: A. 28/3

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

54. Q54. Given that log₂ 16 = 4 and log₁₆ 1048576 = 5, what is log₂ 1048576?

    Free · correct letter only

    A9B20C4/5D1024E5

    Answer

    Answer: B. 20

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

55. Q55. How many integers n > 1 satisfy log₂(n) + log₂(n + 2) < 6?

    Free · correct letter only

    A4B5C6D7E29

    Answer

    Answer: C. 6

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

56. Q56. Given that 9^a = 32 and 2^b = 27, find the value of ab.

    Free · correct letter only

    A3B5/2C6D15/2E15

    Answer

    Answer: D. 15/2

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

57. Q57. Find the greatest value of x + y, where x and y are positive and satisfy log₂(x² / y) = 8 and 4 − log₂(x) = (log₂ x)(log₂ y).

    Free · correct letter only

    A16B1C17D(1 + 256√2) / 512E4

    Answer

    Answer: C. 17

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

58. Q58. The curve y = a x² + 5 passes through (3, log₂(p)) and (log₂(p) − 2, 17), where p is positive. What is p?

    Free · correct letter only

    A8B256C64D16E6

    Answer

    Answer: B. 256

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

59. Q59. Which of the following numbers is the largest?

    Free · correct letter only

    Alog₂(3)B3/2C√2D7/5E8/5

    Answer

    Answer: E. 8/5

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

60. Q60. The equation (log₃(9x))² − 2 log₃(r x) = 0, with r positive and r ≠ 1, has a repeated root. What is r?

    Free · correct letter only

    A3√3B√3C9D3E27

    Answer

    Answer: A. 3√3

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

61. Q61. Find the non-zero solution of 2^36x / 32^6x = 1/16.

    Free · correct letter only

    A1B4Clog base 6 of 4Dlog base 4 of 6E2 times log base 6 of 4

    Answer

    Answer: C. log base 6 of 4

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

62. Q62. You are given that the logarithm base a of (xy) equals 5, the logarithm base a of (yz) equals 4, and the logarithm base a of (xz) equals 3, where x, y and z are real numbers greater than 1. What is a?

    Free · correct letter only

    A1BxCyDzExy

    Answer

    Answer: D. z

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

63. Q63. How many distinct real solutions does log base (x² − 3x + 1) of (x − 2)² = 2 have?

    Free · correct letter only

    A0B1C2D3E4

    Answer

    Answer: B. 1

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

64. Q64. The positive real numbers a and b satisfy 3^a = 32 and 2^b = 243. What is the product ab?

    Free · correct letter only

    A1B5C15D10E25

    Answer

    Answer: E. 25

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

65. Q65. Find the product of the real solutions of (log base 10 of (x²))² + 4 log base 10 of x = 3.

    Free · correct letter only

    A−3/4B10^1/2C10^−1/2D10^−3/2E10⁻¹

    Answer

    Answer: E. 10⁻¹

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

66. Q66. The number a is a valid logarithm base, y is positive, log base a of y = 1/2, and log base 9 of a = x + 2. Which expression equals y?

    Free · correct letter only

    Ax^1/2B9^{x + 2}C3^{(x + 2)/2}D3^{x + 2}E3^{4x + 8}

    Answer

    Answer: D. 3^{x + 2}

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

67. Q67. Throughout this question, log means logarithm to one fixed base. Find x if x log 8 + log 15 = log 10 + x log 27.

    Free · correct letter only

    Alog 3 / log 2B1/3C3D−1/3E1/2

    Answer

    Answer: B. 1/3

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

68. Q68. Positive numbers x and y satisfy log base y of x = 3 and log base 2 of x = 1 + log base 2 of y. What is x + y?

    Free · correct letter only

    A√2B2C2√2D3√2E10

    Answer

    Answer: D. 3√2

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

69. Q69. The numbers x and y satisfy log base 2 of (y − 1) = 2 + log base 2 of x and 2 log base 5 of y = 2 + log base 5 of x. For the solution with the smaller value of x, what is x + y?

    Free · correct letter only

    A1/16B17/16C21/16D5/4E6

    Answer

    Answer: C. 21/16

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

70. Q70. Find the sum of the real solutions of log base 2 of x = 2 + 3 / (log base 2 of x).

    Free · correct letter only

    A2B1/2C17/8D8E17/2

    Answer

    Answer: E. 17/2

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

71. Q71. The power 8^{2x + 1} can be rewritten as 32^a. Express a in terms of x.

    Free · correct letter only

    A(6x + 3)/5B(10x + 5)/3C2x + 1D(2x + 1)/4E(6x + 1)/5

    Answer

    Answer: A. (6x + 3)/5

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

72. Q72. The product (log base 2 of 3)(log base 3 of 4)(log base 4 of 5) ... (log base 31 of 32) is equal to which value?

    Free · correct letter only

    A30B5C1Dlog base 2 of 31E32

    Answer

    Answer: B. 5

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

73. Q73. The numbers log base 10 of 3, log base 10 of (3^x − 1) and log base 10 of (3^x + 5) are three consecutive terms of an arithmetic progression. What is x?

    Free · correct letter only

    A7B3⁷Clog base 7 of 3Dlog base 3 of 7Elog base 10 of 7

    Answer

    Answer: D. log base 3 of 7

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

74. Q74. Positive real numbers a and b satisfy log base 2 of (4a) − log base 2 of b = 3 and log base 2 of a + log base 2 of (2b) = 5. What is a + b?

    Free · correct letter only

    A2√2B4√2C6√2D8E16

    Answer

    Answer: C. 6√2

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

75. Q75. The numbers a and b are nonzero reals. How many positive solutions x does log base 2 of x + log base 2 of (x + a) = b have?

    Free · correct letter only

    Azero if ab < 0, or one if ab > 0Btwo for all nonzero a and bCone for all nonzero a and bDtwo if a < 0, or one if a > 0Ezero if b < 0, or one if b > 0

    Answer

    Answer: C. one for all nonzero a and b

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

76. Q76. Suppose a, b and c are positive and none of them equals 1. They satisfy log base a of b = c, log base b of a = c + 4, and log base c of a = −b. Which statement is true?

    Free · correct letter only

    AThere are infinitely many such triples (a, b, c).BThe value of c is fixed, but a and b are not.CNo such triple exists.DThere is exactly one such triple.EThere are exactly two such triples.

    Answer

    Answer: E. There are exactly two such triples.

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

77. Q77. Let a and b be valid logarithm bases, and let c > 0. The equation log base b of ((b^x)^x) + log base a of (b^x) + (log base a of b)(log base a of c) = 0 has a repeated root when which condition holds?

    Free · correct letter only

    Ab² = 4acBb = c⁴Cb = 1/c⁴Db = 4cEc = b⁴

    Answer

    Answer: B. b = c⁴

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

78. Q78. The real numbers p, q and r are pairwise distinct, and the positive numbers a, b and c satisfy log a / (q − r) = log b / (r − p) = log c / (p − q), with every logarithm taken to the same base. Which of the following must equal 1?

    Free · correct letter only

    Aa^p × b^q × c^rBa^q × b^r × c^pCa^{q − r} × b^{r − p} × c^{p − q}Da^qr × b^rp × c^pqEa^r × b^p × c^q

    Answer

    Answer: A. a^p × b^q × c^r

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

79. Q79. How many real solutions does 8^x − 4 × 4^x − 2^x + 4 = 0 have?

    Free · correct letter only

    A1B2C3D4E0

    Answer

    Answer: B. 2

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

80. Q80. How many real solutions does ln(x² − 5x − 14) = ln(3x + 6) have?

    Free · correct letter only

    A0B1C2D3E4

    Answer

    Answer: B. 1

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

81. Q81. Let f(x) = ln((x + 3)²) and g(x) = 2 ln(x + 3), each defined for the real x at which its formula makes sense. Which statement is true?

    Free · correct letter only

    Af(x) = g(x) for every real x at which f is defined.Bf and g have the same domain.Cf(x) = g(x) for every x > −3, and f(−5) is defined but g(−5) is not.Df(x) = 0 has exactly one real solution.Eg(x) = 0 has two real solutions.

    Answer

    Answer: C. f(x) = g(x) for every x > −3, and f(−5) is defined but g(−5) is not.

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

82. Q82. What are all the real solutions of ln(x² − 2x − 8) = ln(4 − x)?

    Free · correct letter only

    Ax = −3Bx = 4Cx = −3 and x = 4Dx = −2EThere is no real solution.

    Answer

    Answer: A. x = −3

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

83. Q83. How many integers k with −10 ≤ k ≤ 10 make log(kx) = log(x + 1) + log(x + 2) have exactly one real solution?

    Free · correct letter only

    A10B16C15D20E21

    Answer

    Answer: A. 10

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

Paper 2 questions

Paper 2 is Mathematical Reasoning. It can test this same topic. Argument, proof, and identifying errors are the topics that appear on Paper 2 only.

1. Q1. Which set of points (x, y) in the plane satisfies √(x − 2) × ln(x² + y² − 4) = 0?

   Free · correct letter only

   Athe circle x² + y² = 5Bthe line x = 2 except the point (2, 0), together with the part of the circle x² + y² = 5 on which x ≥ 2Cthe whole line x = 2Donly the part of the circle x² + y² = 5 on which x ≥ 2Ethe circle x² + y² = 4

   Answer

   Answer: B. the line x = 2 except the point (2, 0), together with the part of the circle x² + y² = 5 on which x ≥ 2

   The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

2. Q2. What is the sum of all real solutions of log₂(x) + log_x(4) = 3?

   Free · correct letter only

   A6B2C3D4E8

   Answer

   Answer: A. 6

   The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

3. Q3. For which real values of x is 2^{x + 1} > 3^x?

   Free · correct letter only

   Ax > log_3/2 2Bx < log₂(3/2)Cx < log_3/2 2Dx < 1/ln(3/2)Ex < log_2/3 2

   Answer

   Answer: C. x < log_3/2 2

   The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

4. Q4. Which one of the following numbers is smallest in value?

   Free · correct letter only

   Alog base 2 of 6B(2⁻¹ + 2⁻³)⁻¹C2^3/2D1/(√2 − 1)E4 sin²(π/6)

   Answer

   Answer: E. 4 sin²(π/6)

   The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

5. Q5. Consider the equation 3^x = mx + c, where m and c are real constants. Which of the following is true? I: If m < 0, the equation has exactly one real solution for every real c. II: If m = 0, the equation has a real solution if and only if c > 0. III: If m > 0 and c = 0, the equation has a real solution if and only if m > e ln 3.

   Free · correct letter only

   ANone of themBI onlyCII onlyDI and II onlyEI, II and III

   Answer

   Answer: D. I and II only

   The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

6. Q6. Three real numbers x, y and z satisfy x > y > z > 0. Which one of the following must be true?

   Free · correct letter only

   A3^x − 3^y > 3^y − 3^zB3^x + 3^z > 2 × 3^yC(3^x − 3^y)/(x − y) > (3^y − 3^z)/(y − z)D3^x × 3^z > 9^yE(3^x − 3^z)/(x − z) < (3^y − 3^z)/(y − z)

   Answer

   Answer: C. (3^x − 3^y)/(x − y) > (3^y − 3^z)/(y − z)

   The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

7. Q7. The numbers a, b and c are each greater than 1. The following logarithms are all to the same base: log(abc) = 6, log(a²b) = 4 and log(ac) = 4. What is this base?

   Free · correct letter only

   AbBcCThere is not enough information to determine the baseDThe base can be determined, but it is not a, b or cEa

   Answer

   Answer: E. a

   The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

8. Q8. Consider the simultaneous equations p × 3^x + log₃(y) = 4 and 9^x + log₃(y) = 1, where p is real. What is the complete range of p for which there is a real solution (x, y)?

   Free · correct letter only

   Ap > 1Bp > 0Cp ≥ 2√3Dp > 2√3Ep ≤ −2√3 or p ≥ 2√3

   Answer

   Answer: C. p ≥ 2√3

   The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

9. Q9. For x > 1, let f(x) = log₅((log₅ x)/25) and g(x) = log₅(√(log₅ x)). Which statement is true for every x > 1?

   Free · correct letter only

   A0 ≤ f(x) ≤ g(x), or g(x) ≤ f(x) ≤ 0B0 ≤ g(x) ≤ f(x), or f(x) ≤ g(x) ≤ 0C1 ≤ g(x) ≤ f(x), or f(x) ≤ g(x) ≤ 1D2 ≤ g(x) ≤ f(x), or f(x) ≤ g(x) ≤ 2E2 ≤ f(x) ≤ g(x), or g(x) ≤ f(x) ≤ 2

   Answer

   Answer: D. 2 ≤ g(x) ≤ f(x), or f(x) ≤ g(x) ≤ 2

   The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

10. Q10. The numbers a, b and c are greater than 1 and satisfy log base a of b = c and log base b of c = a². What is log base c of a?

    Free · correct letter only

    Aa² cB1/(ac)C1/(a² + c)Da²/cE1/(a² c)

    Answer

    Answer: E. 1/(a² c)

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

11. Q11. Consider the claim that x^y/2 = y^x/2 for all positive real numbers x and y. Which of the following is a counterexample? I: x = 2, y = 4. II: x = 2, y = 6. III: x = 4, y = 2.

    Free · correct letter only

    Anone of themBI onlyCII onlyDIII onlyEI and II only

    Answer

    Answer: C. II only

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

12. Q12. Let k be a real number, and let (*) be the equation 2^x = x + k. Which of the following must have the same number of real solutions as (*)? I: 2^{x − k} = x. II: log₂(x + k) = x. III: 2^x = x − k.

    Free · correct letter only

    AI and III onlyBnone of themCI onlyDIII onlyEI and II only

    Answer

    Answer: E. I and II only

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

13. Q13. Using the observation that 5³ is approximately 11², it is possible to deduce that log₁₁(5) is approximately

    Free · correct letter only

    A2/3B3/2C2D5/11E6

    Answer

    Answer: A. 2/3

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

14. Q14. Each of the numbers below is a root, a power or a logarithm. Which of them is the largest?

    Free · correct letter only

    A2^2/3B5^1/3Clog₈(32)D√3E7^1/4

    Answer

    Answer: D. √3

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

15. Q15. The simultaneous equations 3^2x − 9 × 3^2y = 45 and 3^x + 3 × 3^y = 15 have the solution x = a, y = b. Find a + b.

    Free · correct letter only

    Alog₃(18)Blog₃(11)Clog₃(9/2)D2Elog₃(45)

    Answer

    Answer: A. log₃(18)

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

16. Q16. The numbers x, y and z are positive, and every logarithm below is defined. The equations log_z(x) = y/2, log_x(z) = 8y and log_y(z) = 2 have

    Free · correct letter only

    Ano solutionsBa unique value of y, and infinitely many pairs (x, z)Cunique values of y and z, and infinitely many values of xDtwo different triples (x, y, z)Ea unique triple (x, y, z)

    Answer

    Answer: E. a unique triple (x, y, z)

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

17. Q17. Given that log₂(x) = 3, what is log₄(x)?

    Free · correct letter only

    A3/4B1C3/2D3E6

    Answer

    Answer: C. 3/2

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

18. Q18. Let a, b and c be positive real numbers. Suppose that the logarithm base b of a equals 2, the logarithm base a of (3c − 2) equals 2, and the logarithm base b of (c + 1) equals 2. Which statement about these three conditions is true?

    Free · correct letter only

    AThey specify a uniquelyBThey are satisfied by two values of aCThey are satisfied by infinitely many values of aDThey are contradictoryEThey specify c uniquely but do not specify a

    Answer

    Answer: D. They are contradictory

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

19. Q19. The simultaneous equations 3^2x − 5 × 3^2y = 16 and 3^x + 3^y = 8 have the solution x = a, y = b. What is a + b?

    Free · correct letter only

    Alog base 3 of 2Blog base 3 of 6Clog base 3 of 8Dlog base 3 of 12Elog base 3 of 15

    Answer

    Answer: D. log base 3 of 12

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

20. Q20. The numbers x, y and z are positive, and the logarithms below are defined. The equations log base y of x = 2, log base x of z = y, and log base y of z = 4 have which of the following?

    Free · correct letter only

    Ano solutionsBa unique solution for x, y and zCa unique solution for x and y, but infinitely many solutions for zDa unique solution for y, but infinitely many solutions for x and zEinfinitely many solutions for x, y and z

    Answer

    Answer: B. a unique solution for x, y and z

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

21. Q21. Suppose that a, b and c are real numbers with 1 < a < b and c > 1. Which statement must be true?

    Free · correct letter only

    Alog base a of c is greater than log base b of cBlog base a of c is less than log base b of cClog base a of c equals log base b of cDa^c is greater than b^cEthe information does not determine which of log base a of c and log base b of c is larger

    Answer

    Answer: A. log base a of c is greater than log base b of c

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank

22. Q22. For which values of k does e^2x − 4e^x + k = 0 have exactly two real solutions?

    Free · correct letter only

    Ak < 4B0 < k ≤ 4Ck > 4D0 < k < 4E0 ≤ k < 4

    Answer

    Answer: D. 0 < k < 4

    The worked solution is part of the full bank. The question and the correct letter are free. Unlock the full bank