Logarithms Quiz

Grade 11 · Math · Quiz

10 questions covering log properties, change of base, and solving logarithmic equations.

Questions

1. What is log₁₀(1000)?

   1. 2
   2. 3
   3. 4
   4. 10
   Show answer

   Answer: 3

   10^3 = 1000, so log₁₀(1000) = 3.

2. Simplify: log₆(6^9).

   1. 6
   2. 9
   3. 54
   4. 1
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   Answer: 9

   By the identity log_b(b^n) = n, log₆(6^9) = 9.

3. Which expression equals log₃(x²/y)?

   1. 2log₃(x) − log₃(y)
   2. 2log₃(x) + log₃(y)
   3. log₃(x²) · log₃(y)
   4. log₃(2x − y)
   Show answer

   Answer: 2log₃(x) − log₃(y)

   Power rule gives log₃(x²) = 2log₃(x); quotient rule gives − log₃(y).

4. Solve: log₂(x) = 6.

   1. 12
   2. 32
   3. 64
   4. 36
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   Answer: 64

   2^6 = 64.

5. What is the value of ln(e⁴)?

   1. e
   2. 4e
   3. 4
   4. e⁴
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   Answer: 4

   ln(e^n) = n, so ln(e⁴) = 4.

6. log₄(64) = ?

   1. 2
   2. 3
   3. 4
   4. 16
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   Answer: 3

   4^3 = 64, so log₄(64) = 3.

7. Using change-of-base, log₇(100) ≈ ?

   1. 2.37
   2. 1.43
   3. 0.95
   4. 2.00
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   Answer: 2.37

   log₇(100) = log(100)/log(7) = 2 / 0.845 ≈ 2.37.

8. Solve: log(3x − 2) = 2 (base 10).

   1. x = 34
   2. x = 32
   3. x = 33
   4. x = 36
   Show answer

   Answer: x = 34

   10^2 = 100, so 3x − 2 = 100 → 3x = 102 → x = 34.

9. Which of the following is true?

   1. log(0) = 1
   2. log_b(1) = 0 for any valid base b
   3. log(−5) is a real number
   4. log_b(b²) = b
   Show answer

   Answer: log_b(1) = 0 for any valid base b

   b^0 = 1 for any b > 0, b ≠ 1, so log_b(1) = 0 always.

10. Expand: log₂(8x³).

    1. 3 + 3log₂(x)
    2. 3log₂(8x)
    3. 8 + 3log₂(x)
    4. 3 · log₂(x³)
    Show answer

    Answer: 3 + 3log₂(x)

    log₂(8) = 3 (since 2^3=8); log₂(x³) = 3log₂(x). Sum = 3 + 3log₂(x).

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