Theorems/Properties of logarithms MCQs With Answer

Theorems/Properties of Logarithms MCQs With Answer — This concise, Student-friendly post helps B. Pharm students master logarithms and their theorems for pharmacy calculations. Focused on properties of logarithms, logarithmic identities, change of base, product, quotient and power rules, and natural vs common logs, this guide links core log rules to pharmaceutical topics like pH, pKa and concentration calculations. Suitable for B. Pharm exam prep, these log rules and illustrative MCQs strengthen analytical skills and numerical accuracy required in pharmacokinetics and formulation math. Clear examples and applied problems make learning practical and exam-ready. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. Which property explains that log_a(xy) = log_a(x) + log_a(y)?

• Product rule of logarithms
• Quotient rule of logarithms
• Power rule of logarithms
• Change of base rule

Correct Answer: Product rule of logarithms

Q2. Simplify log10(1000) using properties of logarithms.

• 2
• 3
• 4
• 1

Correct Answer: 3

Q3. Which identity is used to convert log_a(b) to natural logs?

• log_a(b) = ln(b) / ln(a)
• log_a(b) = ln(a) / ln(b)
• log_a(b) = ln(a*b)
• log_a(b) = ln(b – a)

Correct Answer: log_a(b) = ln(b) / ln(a)

Q4. If log10(x) = 2.5, what is x?

• 316.23
• 25
• 2.5
• 158.11

Correct Answer: 316.23

Q5. Using log rules, simplify log_a(x^3).

• 3 log_a(x)
• log_a(x) / 3
• log_a(3x)
• log_a(x)^3

Correct Answer: 3 log_a(x)

Q6. Which formula relates pH to hydrogen ion concentration [H+]? (Common in pharm calculations)

• pH = -log10([H+])
• pH = log10([H+])
• pH = ln([H+])
• pH = -ln([H+])

Correct Answer: pH = -log10([H+])

Q7. Using properties, log10(0.01) equals:

• -2
• 2
• -0.01
• 0.01

Correct Answer: -2

Q8. Which property gives log_a(x/y) = log_a(x) – log_a(y)?

• Quotient rule
• Product rule
• Power rule
• Reciprocal rule

Correct Answer: Quotient rule

Q9. If ln(5) ≈ 1.609 and ln(2) ≈ 0.693, what is ln(5/2)?

• 0.916
• 2.302
• 0.500
• 1.916

Correct Answer: 0.916

Q10. Change of base: log2(32) equals:

• 5
• 4
• 2
• 10

Correct Answer: 5

Q11. Simplify using logs: log10(2) + log10(50) equals:

• log10(100)
• log10(48)
• log10(52)
• log10(0.25)

Correct Answer: log10(100)

Q12. Evaluate: log10(100) – log10(4) equals:

• log10(25)
• log10(400)
• log10(96)
• log10(0.25)

Correct Answer: log10(25)

Q13. Which property allows moving exponents forward: log_a(b^c) = c log_a(b)?

• Power property
• Product property
• Change of base
• Inverse property

Correct Answer: Power property

Q14. Solve for x: log10(x) = -3. What is x?

• 0.001
• 1000
• -0.001
• 3

Correct Answer: 0.001

Q15. Which is true about logs and reciprocals: log_a(1/x) equals?

• -log_a(x)
• log_a(x)
• 1/log_a(x)
• log_a(-x)

Correct Answer: -log_a(x)

Q16. For pharmacokinetics, if concentration C doubles, how does log10(C) change? (Assume base-10)

• Increases by log10(2)
• Doubles
• Increases by 2
• Remains same

Correct Answer: Increases by log10(2)

Q17. Convert to natural log: log10(8) equals which using ln?

• ln(8)/ln(10)
• ln(10)/ln(8)
• ln(8)*ln(10)
• ln(8) – ln(10)

Correct Answer: ln(8)/ln(10)

Q18. If log_a(b) = 2 and a = 3, what is b?

• 9
• 6
• 3^2 = 6
• 1/9

Correct Answer: 9

Q19. Which statement is correct for base conversion? log_b(a) = 1 / log_a(b) true or false?

• True
• False
• Only for base 10
• Only for natural log

Correct Answer: True

Q20. Using log rules, simplify log10(0.5) + log10(200).

• log10(100)
• log10(400)
• log10(0.25)
• log10(250)

Correct Answer: log10(100)

Q21. Solve for x: ln(x) = 4.5. Which is x? (Use e^y relation)

• e^4.5
• 4.5
• ln(4.5)
• 10^4.5

Correct Answer: e^4.5

Q22. If pKa = 4.76 and acid fraction requires pH = pKa + log([A-]/[HA]), which log is used?

• Log10
• Natural log (ln)
• Log base 2
• Any base without change

Correct Answer: Log10

Q23. Which expression equals log_a(a^x)?

• x
• a^x
• log_a(x)
• 1/x

Correct Answer: x

Q24. Evaluate: log10(3) + log10(7) equals:

• log10(21)
• log10(10)
• log10(0.4286)
• log10(4)

Correct Answer: log10(21)

Q25. Which of the following is NOT a property of logarithms?

• log_a(x + y) = log_a(x) + log_a(y)
• log_a(xy) = log_a(x) + log_a(y)
• log_a(x/y) = log_a(x) – log_a(y)
• log_a(x^k) = k log_a(x)

Correct Answer: log_a(x + y) = log_a(x) + log_a(y)

Q26. Solve for x: log2(x) = 3.5. Which equals x?

• 2^3.5
• 3.5
• 10^3.5
• 2 * 3.5

Correct Answer: 2^3.5

Q27. If ln(A) = 2 and ln(B) = 3, what is ln(A^2 * B)?

• 2*2 + 3 = 7
• 2 + 3 = 5
• 4 + 3 = 7
• 2^2 * 3 = 7

Correct Answer: 4 + 3 = 7

Q28. In pharmacology logs, which is correct: log10(AB) = ?

• log10(A) + log10(B)
• log10(A) – log10(B)
• log10(A) * log10(B)
• log10(A^B)

Correct Answer: log10(A) + log10(B)

Q29. Use properties: log10(250) – log10(2) simplifies to:

• log10(125)
• log10(500)
• log10(248)
• log10(125000)

Correct Answer: log10(125)

Q30. If log_a(b) = m and log_a(c) = n, then log_a(b^2 c) equals:

• 2m + n
• m + 2n
• mn
• m^2 + n

Correct Answer: 2m + n

Q31. Which property helps approximate drug concentration changes on a logarithmic scale?

• Logarithmic additivity for ratios
• Logarithm of sum property
• Logarithm of negative numbers
• Logarithm integration rule

Correct Answer: Logarithmic additivity for ratios

Q32. Compute: log10(4) + log10(25) equals:

• log10(100)
• log10(29)
• log10(6.25)
• log10(10)

Correct Answer: log10(100)

Q33. Which equality is true for any positive x and base a>0, a≠1?

• a^(log_a(x)) = x
• log_a(a + x) = x
• log_a(x) = a^x
• a^(log_a x) = log_a x

Correct Answer: a^(log_a(x)) = x

Q34. If pH change from 7 to 6, how does [H+] change approximately? (Using log relation)

• Increases 10-fold
• Decreases 10-fold
• Remains unchanged
• Increases 2-fold

Correct Answer: Increases 10-fold

Q35. Simplify: log10(81) using base conversion if 81 = 3^4; express as log10(3^4).

• 4 log10(3)
• log10(3^4) = log10(3)/4
• log10(3)^4
• log10(3) + 4

Correct Answer: 4 log10(3)

Q36. Which expression equals log_a(b) * log_b(a)?

• 1
• log_a(a)
• log_b(b)
• 0

Correct Answer: 1

Q37. Solve: If log10(x) = 1/2, what is x?

• √10
• 10^2
• 0.5
• 2

Correct Answer: √10

Q38. Which is correct for combining logs: log_a(x^y) equals?

• y log_a(x)
• log_a(y) + log_a(x)
• log_a(x) / y
• log_a(x^y) = log_a(x) * log_a(y)

Correct Answer: y log_a(x)

Q39. If ln(10) ≈ 2.3026, what is log10(e)?

• 1 / ln(10) ≈ 0.4343
• ln(10) ≈ 2.3026
• e
• 10

Correct Answer: 1 / ln(10) ≈ 0.4343

Q40. For solving chemical kinetics, which log property helps linearize exponential decay?

• Taking ln of both sides to convert exponent to multiplication
• Using product rule for sums
• Using log of a sum
• Changing base to 2

Correct Answer: Taking ln of both sides to convert exponent to multiplication

Q41. Evaluate: log10(0.2) + log10(5) equals:

• log10(1)
• log10(25)
• log10(0.04)
• log10(1.0)

Correct Answer: log10(1)

Q42. Which is the inverse relationship of logarithm?

• Exponential function
• Square function
• Sine function
• Reciprocal function

Correct Answer: Exponential function

Q43. Solve: log5(125) equals:

• 3
• 5
• 1/3
• 25

Correct Answer: 3

Q44. Which property allows splitting ln(xy/z) into a sum and difference?

• ln(xy/z) = ln(x) + ln(y) – ln(z)
• ln(xy/z) = ln(x + y) – ln(z)
• ln(xy/z) = ln(x) * ln(y) / ln(z)
• ln(xy/z) = ln(xy) / ln(z)

Correct Answer: ln(xy/z) = ln(x) + ln(y) – ln(z)

Q45. If log10(a) = p and log10(b) = q, what is log10(a^2 b^3)?

• 2p + 3q
• p^2 + q^3
• p + q
• 2pq + 3

Correct Answer: 2p + 3q

Q46. True or false: log_a(-x) is defined for positive base a and positive x.

• False
• True
• Only if a = 10
• Only if x = 1

Correct Answer: False

Q47. Solve for x: log10(3x) = 2. What is x?

• 100/3
• 33
• 10/3
• 300

Correct Answer: 100/3

Q48. Which of these is a useful approach when solving log equations with different bases?

• Convert to a common base using change-of-base
• Square both sides immediately
• Add the logs directly without rules
• Ignore the base and proceed

Correct Answer: Convert to a common base using change-of-base

Q49. For dilution calculations, if concentration reduces by factor of 100, how does log10(C) change?

• Decreases by 2
• Increases by 2
• Decreases by 100
• Remains same

Correct Answer: Decreases by 2

Q50. Which statement is true: log_a(1) equals?

• 0
• 1
• a
• Undefined

Correct Answer: 0

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