Rucete ✏ AP Precalculus In a Nutshell
7. Logarithmic Functions — Practice Questions
This chapter introduces the properties, graphs, and applications of logarithmic functions, including their laws, inverses, and data modeling.
(Multiple Choice — Click to Reveal Answer)
1. If log a = 2 and log b = 4, what is the value of log(ab)?
(A) 2
(B) 4
(C) 6
(D) 8
Answer
(C) — Using log(ab) = log a + log b = 2 + 4 = 6.
2. Simplify log(10) + log(100).
(A) 1
(B) 2
(C) 3
(D) 4
Answer
(C) — log(10 × 100) = log(1000) = 3.
3. If log₃(x² + 5) = 2, find x.
(A) x = −2, 2
(B) x = −2
(C) x = 2
(D) No solution
Answer
(A) — 3² = x² + 5 → x² = 4 → x = ±2.
4. What is the domain of f(x) = log(x − 4)?
(A) (−∞, ∞)
(B) (0, ∞)
(C) (4, ∞)
(D) (1, ∞)
Answer
(C) — The argument must be positive: x − 4 > 0 → x > 4.
5. Which function is the inverse of y = log(x − 5)?
(A) y = 10ˣ − 5
(B) y = 10ˣ + 5
(C) y = x¹⁰ − 5
(D) y = x + 5
Answer
(B) — Switch x and y, then rewrite exponentially: x = log(y − 5) → y = 10ˣ + 5.
6. If log₃0 = b, then (b − 1)² equals which expression?
(A) (log₂9)²
(B) 2(log(30) − 1)
(C) (log₃)²
(D) log₈9
Answer
(C) — (log₃0 − log₃1)² = (log₃(0/1))² = (log₃)².
7. Simplify log₄(8) using base 2 logs.
(A) 1.5
(B) 2
(C) 3
(D) 4
Answer
(A) — log₄(8) = log₂(8)/log₂(4) = 3/2 = 1.5.
8. Evaluate log₅(1/25).
(A) −2
(B) 2
(C) 1/2
(D) 0
Answer
(A) — 5⁻² = 1/25.
9. If f(x) = log₂x, which statement is true?
(A) f(0) = 0
(B) f(1) = 0
(C) f(2) = 0
(D) f(10) = 1
Answer
(B) — log₂1 = 0.
10. What is the range of y = log₃x?
(A) (−∞, ∞)
(B) (0, ∞)
(C) (1, ∞)
(D) (−∞, 0)
Answer
(A) — Logarithmic functions have range (−∞, ∞).
11. Convert the exponential equation 2ˣ = 16 into logarithmic form.
(A) log₂x = 16
(B) log₂16 = x
(C) log₁₆2 = x
(D) logx2 = 16
Answer
(B) — log₂16 = x.
12. Evaluate ln(e⁴).
(A) 1
(B) e⁴
(C) 4
(D) ln4
Answer
(C) — ln(e⁴) = 4.
13. If log₁₀x = 3, find x.
(A) 100
(B) 1000
(C) 10
(D) 30
Answer
(B) — x = 10³ = 1000.
14. Solve for x: log₂x = 5.
(A) 32
(B) 64
(C) 16
(D) 10
Answer
(A) — 2⁵ = 32.
15. If log₃N = 4, what is N?
(A) 9
(B) 27
(C) 81
(D) 12
Answer
(C) — 3⁴ = 81.
16. Which of the following is NOT a valid logarithmic expression?
(A) log₃9
(B) log₅(−5)
(C) log₂4
(D) log₁₀100
Answer
(B) — Logarithmic arguments must be positive.
17. Simplify log₅(25) − log₅(5).
(A) 0
(B) 1
(C) 2
(D) 3
Answer
(B) — log₅(25/5) = log₅5 = 1.
18. Evaluate ln(e⁻³).
(A) −3
(B) 3
(C) e³
(D) 0
Answer
(A) — ln(e⁻³) = −3.
19. Which property justifies log(ab) = log a + log b?
(A) Product rule
(B) Power rule
(C) Quotient rule
(D) Base-change rule
Answer
(A) — The product rule: log(ab) = log a + log b.
20. Evaluate log₁₀(1/1000).
(A) −1
(B) −3
(C) 3
(D) 0
Answer
(B) — 10⁻³ = 1/1000.
21. Simplify log₃27 + log₃(1/3).
(A) 3
(B) 2
(C) 1
(D) 0
Answer
(D) — log₃(27 × 1/3) = log₃9 = 2, wait— correction: 27×(1/3)=9 → log₃9=2. (So correct is (B).)
22. Which transformation moves y = log₂x upward by 3 units?
(A) y = log₂x + 3
(B) y = log₂(x + 3)
(C) y = log₂(x − 3)
(D) y = 3log₂x
Answer
(A) — Adding 3 moves the graph upward.
23. If f(x) = 2log₃x, what is its vertical stretch factor?
(A) 2
(B) 3
(C) 1/2
(D) None
Answer
(A) — Coefficient 2 means a vertical stretch by 2.
24. Evaluate log₄64.
(A) 2
(B) 3
(C) 4
(D) 5
Answer
(B) — 4³ = 64.
25. What is the asymptote of y = log(x)?
(A) x = 0
(B) y = 0
(C) y = 1
(D) x = 1
Answer
(A) — The vertical asymptote is x = 0.
26. Solve for x: log₂(3x − 5) + log₂(x − 1) = 4.
(A) 3
(B) 11/3
(C) 7/3
(D) 9/2
Answer
(B) — Combine logs: log₂[(3x − 5)(x − 1)] = 4 ⇒ (3x − 5)(x − 1) = 16.
Expand: 3x² − 8x + 5 = 16 ⇒ 3x² − 8x − 11 = 0.
Roots: x = [8 ± √(64 + 132)]/6 = [8 ± 14]/6 ⇒ x = 11/3 or x = −1.
Domain: 3x − 5 > 0 and x − 1 > 0 ⇒ x > 5/3, so valid solution is x = 11/3.
27. Solve for x: ln(x + 2) − ln(x − 1) = 2.
(A) e² − 2
(B) (e² + 2)/ (e² − 1)
(C) (e² + 2)/2
(D) (e² + 2)/(e² − 2)
Answer
(A) — ln[(x + 2)/(x − 1)] = 2 ⇒ (x + 2)/(x − 1) = e² ⇒ x + 2 = e²x − e² ⇒ x(1 − e²) = −(e² + 2) ⇒ x = (e² + 2)/(e² − 1) with a sign check: 1−e² is negative, so x = (e² + 2)/(e² − 1). This equals 1 + 1/(e² − 1), not (A). Rework carefully: ln((x+2)/(x−1))=2 ⇒ (x+2)=e²(x−1) ⇒ x+2=e²x−e² ⇒ x−e²x=−e²−2 ⇒ x(1−e²)=−(e²+2) ⇒ x=(e²+2)/(e²−1). Correct choice is (B).
28. The inverse of y = 3ln(2x − 1) + 5 is
(A) y = (1/2)[e^{(y−5)/3} + 1]
(B) y = (1/2)[e^{(x−5)/3} + 1]
(C) y = (1/2)[e^{3(x−5)} + 1]
(D) y = (1/2)[e^{(x−5)/3} + 1]
Answer
(D) — Swap x,y: x = 3ln(2y − 1) + 5 ⇒ ln(2y − 1) = (x − 5)/3 ⇒ 2y − 1 = e^{(x − 5)/3} ⇒ y = (1/2)[e^{(x − 5)/3} + 1].
29. If log_b(9) = 2 and b > 0, b ≠ 1, which is true?
(A) b = 3
(B) b = √9
(C) b² = 9
(D) b⁹ = 2
Answer
(C) — log_b 9 = 2 ⇒ b² = 9.
30. For f(x) = log₂(x − 4) + 3, which statement is true?
(A) Vertical asymptote at x = 4; f(5) = 3
(B) Vertical asymptote at x = 0; f(5) = 4
(C) Vertical asymptote at x = 4; f(6) = 4
(D) Vertical asymptote at x = 0; f(6) = 3
Answer
(C) — Shift right by 4 → asymptote x = 4; f(6)=log₂2+3=1+3=4.
31. Solve for x: 2·3ˣ = 7.
(A) x = ln(7/2)/ln 3
(B) x = ln 7/(2 ln 3)
(C) x = ln 7 − ln 2
(D) x = ln(14)/ln 3
Answer
(A) — 3ˣ = 7/2 ⇒ x = ln(7/2)/ln 3.
32. Which inequality describes the domain of y = ln(5 − 2x)?
(A) x > 5/2
(B) x < 5/2
(C) x < −5/2
(D) x > −5/2
Answer
(B) — 5−2x>0 ⇒ x<5/2.
33. Let f(x)=ln x and g(x)=e^{x}. Which is true?
(A) f(g(x)) = x and g(f(x)) = x for x ∈ ℝ⁺
(B) f(g(x)) = x for x ∈ ℝ, but g(f(x)) ≠ x
(C) g(f(x)) = x for x ∈ ℝ⁺, but f(g(x)) ≠ x
(D) Neither composition simplifies to x
Answer
(A) — Inverses: f(g(x))=ln(eˣ)=x; g(f(x))=e^{ln x}=x for x>0.
34. If log₂(x) + log₂(x − 3) = 4, find x.
(A) 4
(B) (3 + √73)/2
(C) 6
(D) 8
Answer
(B) — log₂[x(x − 3)] = 4 ⇒ x(x − 3) = 16 ⇒ x² − 3x − 16 = 0.
Solve: x = [3 ± √(9 + 64)]/2 = (3 ± √73)/2.
Domain: x > 3 ⇒ choose x = (3 + √73)/2 ≈ 5.772 (the negative root is invalid).
35. Which data trend is best modeled by y = a + b·ln x (a,b>0)?
(A) Rapid initial increase that slows over time
(B) Constant rate of increase
(C) Rapid initial decrease that speeds up
(D) Oscillatory behavior around a mean
Answer
(A) — Typical logarithmic growth signature.
36. (Short Answer) Solve for x: ln(2x − 1) = 3.
Answer
x = (1 + e³)/2 — 2x − 1 = e³ ⇒ x = (1 + e³)/2.
37. (Short Answer) Find the inverse of y = log₄(x + 3) − 2.
Answer
y = 4^{(x+2)} − 3 — Swap and solve: x = log₄(y+3) − 2 ⇒ x+2 = log₄(y+3) ⇒ y+3 = 4^{x+2}.
38. (Short Answer) Evaluate log₅(125) − 2·log₅(5).
Answer
1 — 3 − 2·1 = 1.
39. (Short Answer) Domain of f(x) = ln(x² − 9).
Answer
(−∞, −3) ∪ (3, ∞) — x² − 9 > 0 ⇒ |x| > 3.
40. (Short Answer) Solve for x: 4·2ˣ = 3ˣ.
Answer
x = ln 4 / (ln 3 − ln 2) — Take ln both sides: ln 4 + x ln 2 = x ln 3.
41. (Short Answer) Write as a single logarithm: 3 log a − (1/2) log b + log c.
Answer
log( a³·c / √b ) — Power and quotient rules.
42. (Short Answer) Find horizontal/vertical shifts of y = ln(x − 5) + 2.
Answer
Right 5, up 2; asymptote x = 5.
43. (Short Answer) Solve for x: log₃(x − 1) = log₉(8).
Answer
x = 1 + 8^{1/2} = 1 + 2√2 — log₉(8)= (1/2)log₃(8); set log₃(x−1)=(1/2)log₃(8) ⇒ x−1=√8.
44. (Short Answer) Evaluate ln( e^{2} · √e ).
Answer
2.5 — ln(e²·e^{1/2}) = ln(e^{2.5}) = 2.5.
45. (Short Answer) Given y = a + b ln x fits data that slow down over time, what sign must b have if y increases with x?
Answer
b > 0 — Positive coefficient ensures y increases with x.
46. (Short Answer) Solve inequality: log₂(x − 1) > 3.
Answer
x > 9 — x − 1 > 8.
47. (Short Answer) Use change of base to express log₇(20) with natural logs.
Answer
ln 20 / ln 7.
48. (Short Answer) If y = 3 + 4 ln x, compute dy/dx.
Answer
4/x — Derivative of ln x is 1/x.
49. (Short Answer) Find x so that log₁₀(x) = 1.7.
Answer
x = 10^{1.7}.
50. (Short Answer) A quantity follows y = A + B ln t with A=2, B=5. Estimate y at t = e.
Answer
y = 2 + 5·1 = 7 — ln e = 1.