20 Logarithm Quiz Questions and Answers

Question 1:
What is the value of log₁₀(100)?
A) 1
B) 2
C) 10
D) 100

Answer: B) 2

Explanation: log₁₀(100) = 2 because 10 raised to the power of 2 equals 100.

Question 2:
Solve for x: 2^x = 8
A) x = 1
B) x = 2
C) x = 3
D) x = 4

Answer: C) 3

Explanation: 2^3 = 8, so x = 3.

Question 3:
What is log_b(1) for any base b > 0 and b ≠ 1?
A) 0
B) 1
C) b
D) Undefined

Answer: A) 0

Explanation: log_b(1) = 0 because b^0 = 1.

Question 4:
Simplify log(a) + log(b) using logarithm properties.
A) log(a + b)
B) log(ab)
C) log(a – b)
D) log(a/b)

Answer: B) log(ab)

Explanation: The product rule states that log(a) + log(b) = log(ab).

Question 5:
Simplify log(a) – log(b) using logarithm properties.
A) log(a + b)
B) log(ab)
C) log(a/b)
D) log(b/a)

Answer: C) log(a/b)

Explanation: The quotient rule states that log(a) – log(b) = log(a/b).

Question 6:
Simplify n * log(a) using logarithm properties.
A) log(a^n)
B) log(n^a)
C) log(a + n)
D) log(a – n)

Answer: A) log(a^n)

Explanation: The power rule states that n * log(a) = log(a^n).

Question 7:
What is log₂(8) using the change of base formula?
A) log(8) / log(2) where base is 10
B) 2 * log(8)
C) log(2) / log(8)
D) 8 / 2

Answer: A) log(8) / log(2) where base is 10

Explanation: The change of base formula is log_b(a) = log_c(a) / log_c(b), so log₂(8) = log(8) / log(2).

Question 8:
What is the domain of the function y = log(x)?
A) All real numbers
B) x > 0
C) x ≥ 0
D) x < 0 Answer: B) x > 0

Explanation: The argument of a logarithm must be positive, so the domain is x > 0.

Question 9:
Solve for x: log₁₀(x) = 2
A) x = 10
B) x = 100
C) x = 2
D) x = 1

Answer: B) 100

Explanation: log₁₀(x) = 2 means 10^2 = x, so x = 100.

Question 10:
What is the value of log_e(e)?
A) 0
B) 1
C) e
D) Undefined

Answer: B) 1

Explanation: log_e(e) = 1 because e^1 = e.

Question 11:
Simplify log_b(b^x).
A) x
B) b
C) 1
D) b^x

Answer: A) x

Explanation: log_b(b^x) = x by the definition of logarithms.

Question 12:
If log_2(4) = 2, what is log_2(8)?
A) 1
B) 2
C) 3
D) 4

Answer: C) 3

Explanation: 2^3 = 8, so log_2(8) = 3.

Question 13:
Simplify log(1000) where the base is 10.
A) 2
B) 3
C) 4
D) 10

Answer: B) 3

Explanation: log₁₀(1000) = 3 because 10^3 = 1000.

Question 14:
What is the inverse function of y = 2^x?
A) y = log_2(x)
B) y = x^2
C) y = 2x
D) y = e^x

Answer: A) y = log_2(x)

Explanation: The inverse of an exponential function y = a^x is y = log_a(x).

Question 15:
Solve for x: log(x) + log(3) = log(15) (base 10)
A) x = 5
B) x = 12
C) x = 15
D) x = 45

Answer: A) x = 5

Explanation: Using the product rule, log(x * 3) = log(15), so x * 3 = 15, thus x = 5.

Question 16:
What is the value of ln(e^3)?
A) 1
B) 3
C) e
D) e^3

Answer: B) 3

Explanation: ln(e^3) = 3 because the natural logarithm and exponential are inverses.

Question 17:
Simplify log_b(a) * log_a(b).
A) 1
B) b
C) a
D) Undefined

Answer: A) 1

Explanation: log_b(a) * log_a(b) = 1 by the change of base property.

Question 18:
Solve for x: 3^{x+1} = 27
A) x = 2
B) x = 3
C) x = 1
D) x = 0

Answer: A) x = 2

Explanation: 27 = 3^3, so 3^{x+1} = 3^3, thus x + 1 = 3, and x = 2.

Question 19:
What is the range of y = log(x)?
A) All real numbers
B) x > 0
C) y > 0
D) y ≥ 0

Answer: A) All real numbers

Explanation: The logarithmic function y = log(x) can produce any real number as output.

Question 20:
If log_3(9) = 2, what is log_3(27)?
A) 1
B) 2
C) 3
D) 9

Answer: C) 3

Explanation: 3^3 = 27, so log_3(27) = 3.