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Math 30-1 · Worksheets

Exponents and logs · Set B

Ten questions of mixed difficulty, covering Exponents and logs. Print it, or work through it on screen — the answer key starts on its own page.

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Exponents and logs · Set B

Math 30-1 · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. Solve 3ˣ = 20, to the nearest hundredth.

    1. a) 2.73
    2. b) 1.82
    3. c) 6.67
    4. d) 0.37

  2. Evaluate log 1000.

    1. a) 100
    2. b) 3
    3. c) 2
    4. d) 4

  3. Solve 2ˣ = 32.

    1. a) 5
    2. b) 6
    3. c) 16
    4. d) 4

  4. A sample of 80 g has a half-life of 5 years. How much remains after 15 years?

    1. a) 40 g
    2. b) 10 g
    3. c) 20 g
    4. d) 5 g

  5. On the Richter scale, how many times more intense is a magnitude 7 earthquake than a magnitude 5?

    1. a) 20
    2. b) 1000
    3. c) 100
    4. d) 2

  6. What is the horizontal asymptote of y = 2ˣ?

    1. a) y = 1
    2. b) y = 2
    3. c) x = 0
    4. d) y = 0

  7. Solve log₂ x + log₂(x − 2) = 3.

    1. a) x = −2 only
    2. b) x = 4 only
    3. c) x = 8
    4. d) x = 4 and x = −2

  8. Solve 4ˣ⁺¹ = 8ˣ.

    1. a) 1
    2. b) 3
    3. c) 2
    4. d) 4

  9. Evaluate log₃(1/9).

    1. a) −2
    2. b) 1/2
    3. c) 2
    4. d) −3

  10. Evaluate log₂ 12 − log₂ 3.

    1. a) log₂ 9
    2. b) 3
    3. c) 4
    4. d) 2

Answer key · Exponents and logs · Set B

Math 30-1 · maddyhelps.com

  1. a) 2.73 — No common base, so take logs: x log 3 = log 20, giving x = log 20 ÷ log 3 ≈ 2.73. Check: 3² = 9 and 3³ = 27, so it must be between 2 and 3.
  2. b) 3 — With no base written, the base is 10. 10³ = 1000, so log 1000 = 3.
  3. a) 5 — Write 32 as a power of 2: 32 = 2⁵. Same base on both sides means the exponents match, so x = 5.
  4. b) 10 g — 15 years is 3 half-lives: 80 → 40 → 20 → 10 g. In formula form, 80(1/2)^(15/5) = 10.
  5. c) 100 — Each step up is 10 times more intense. Two steps is 10² = 100 times, not 2.
  6. d) y = 0 — As x gets very negative, 2ˣ shrinks toward 0 but never reaches it, so the graph hugs y = 0. y = 1 is the y-intercept.
  7. b) x = 4 only — Combine: log₂[x(x − 2)] = 3, so x² − 2x = 8, giving x = 4 or x = −2. But log₂(−2) does not exist, so −2 is extraneous. Always check.
  8. c) 2 — Rewrite both with base 2: 2²⁽ˣ⁺¹⁾ = 2³ˣ. Match exponents: 2x + 2 = 3x, so x = 2.
  9. a) −2 — 1/9 = 3⁻², so the exponent is −2. A fraction less than 1 always gives a negative log when the base is bigger than 1.
  10. d) 2 — Subtracting logs with the same base means dividing inside: log₂(12 ÷ 3) = log₂ 4 = 2. log₂ 9 wrongly subtracts the insides.