Grade 11 Exponential & Logarithmic Functions Practice — Easy

Question 1 of 28Score 0/0Easy

Question 1 of 28: log_b(x) = y means:

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Answer key for parents & teachers (28 questions)
  1. log_b(x) = y means: — b^y = x. log_b(x) = y is equivalent to b^y = x. Logarithm is the exponent.
  2. log_5(25) = ? — 2. 5² = 25, so log_5(25) = 2.
  3. log(1) = ? (any base) — 0. b⁰ = 1 for any base b, so log_b(1) = 0 for any base.
  4. log_b(b) = ? — 1. b¹ = b, so log_b(b) = 1. A log base b of b is always 1.
  5. The product rule states: log_b(MN) = ? — log_b(M) + log_b(N). log_b(MN) = log_b(M) + log_b(N). Logarithms turn multiplication into addition.
  6. The quotient rule states: log_b(M/N) = ? — log_b(M) − log_b(N). log_b(M/N) = log_b(M) − log_b(N). Division inside a log becomes subtraction.
  7. The power rule states: log_b(Mᵖ) = ? — p · log_b(M). log_b(Mᵖ) = p · log_b(M). The exponent moves in front.
  8. log₂(32) = ? — 5. 2⁵ = 32, so log₂(32) = 5.
  9. ln(e⁴) = ? — 4. ln(e⁴) = 4 · ln(e) = 4 · 1 = 4. (Power rule with natural log.)
  10. e^(ln 7) = ? — 7. e^(ln 7) = 7. Exponential and natural log are inverses.
  11. log₁₀(10000) = ? — 4. 10⁴ = 10000, so log₁₀(10000) = 4.
  12. If log₂(x) = 5, then x = ? — 32. log₂(x) = 5 → x = 2⁵ = 32.
  13. To solve 5^x = 125, you recognize that 125 = 5^? — 3. 5³ = 125. So 5^x = 5³ → x = 3.
  14. Solve: 3^x = 81 — x = 4. 3⁴ = 81, so x = 4.
  15. Solve: 2^(x+1) = 32 — x = 4. 32 = 2⁵, so x + 1 = 5 → x = 4.
  16. To solve 5^x = 7 (where we cannot express 7 as a power of 5), the first step is: — Take log of both sides. Take log (or ln) of both sides: x·log(5) = log(7) → x = log(7)/log(5).
  17. Solve: e^x = 20 — x = ln(20). Take ln of both sides: x = ln(20) ≈ 2.996.
  18. Solve: log₃(x) = 4 — x = 81. log₃(x) = 4 → x = 3⁴ = 81.
  19. Solve: 6^x = 1 — x = 0. Any nonzero base raised to power 0 equals 1. So 6^x = 1 → x = 0.
  20. The value of e is approximately: — 2.71828. e ≈ 2.71828… It is the base of the natural logarithm and arises in continuous growth.
  21. The natural logarithm ln(x) is log base: — e. ln(x) = log_e(x). The natural logarithm uses base e.
  22. In the model A(t) = A₀eʳᵗ, positive r means: — Growth. When r > 0, the exponent rt grows with time, so A(t) increases — this is growth.
  23. In the model A(t) = A₀eʳᵗ, negative r means: — Decay. When r < 0, rt becomes more negative over time, so e^(rt) decreases — this is decay.
  24. If a substance has half-life 5 years, after 10 years the remaining amount is: — 1/4 of original. Two half-lives pass: (1/2)² = 1/4 of the original remains.
  25. ln(e) = ? — 1. ln(e) = log_e(e) = 1. Just as log₁₀(10) = 1.
  26. e^0 = ? — 1. Any number to the power 0 equals 1. e^0 = 1.
  27. Which graph best describes y = e^(−x)? — Starts high, decreases toward y = 0. e^(−x) is a decreasing exponential. As x→∞, e^(−x)→0 (asymptote at y=0). At x=0, y=1.
  28. For A(t) = A₀e^(rt), what is the y-intercept? — A₀. At t = 0: A(0) = A₀e^0 = A₀. The initial value A₀ is the y-intercept.