Key Concepts in Logarithms

Author: Justice

This quiz covers definitions, domain conditions, basic properties, and computational rules for logarithmic functions, including evaluating logarithms and solving simple logarithmic equations.

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1 How is log_a(b) defined in terms of the base a and the number b?

A It is the a-th root taken from the number b B It is the exponent to which a must be raised to obtain b C It is the product obtained by multiplying a and b D It is the quotient obtained by dividing b by a

2 What is the domain of the function f(x) = log_a(x) for an admissible base a?

A All natural numbers greater than or equal to one B All non‑negative real numbers including zero C All positive real numbers only D All real numbers without any restriction

3 Which conditions must the base a of a logarithm satisfy for log_a(x) to be defined?

A a < 0 and a ≠ −1 at the same time B a ≥ 1 and a is an integer number C a > 0 with no additional restriction D a > 0 and a ≠ 1 simultaneously

4 For any admissible base a, what is the value of log_a(1)?

A It is always equal to 0 B It is always equal to 1 C It does not exist for any base D It is always equal to a

5 What is the value of the common logarithm log₁₀(1000)?

A 2 because 10² equals 1000 B 1 because 10¹ equals 1000 C 0 because 10⁰ equals 1000 D 3 because 10³ equals 1000

6 Evaluate the natural logarithm ln(e⁵).

A 5 because ln(e⁵) equals 5·ln(e) B 1 because ln(e⁵) equals ln(e) C e because ln(e⁵) equals e·ln(5) D 0 because ln(e⁵) equals ln(1)

7 Given that 32 = 2⁵, what is the value of log₂(32)?

A 5 because 2 raised to 5 equals 32 B 3 because 2 raised to 3 equals 32 C 4 because 2 raised to 4 equals 32 D 2 because 2 raised to 2 equals 32

8 Compute log₃(9) + log₃(27) using logarithm properties.

A 5 because log₃(9·27) equals log₃(3⁵) B 3 because log₃(9·27) equals log₃(3³) C 2 because log₃(9·27) equals log₃(3²) D 4 because log₃(9·27) equals log₃(3⁴)

9 Find the value of log₅(125) − log₅(5) using logarithm rules.

A 2 because log₅(125/5) equals log₅(25) B 0 because log₅(125/5) equals log₅(1) C 3 because log₅(125/5) equals log₅(125) D 1 because log₅(125/5) equals log₅(5)

10 Evaluate log₂(8⁴) using the power rule for logarithms.

A 12 because 4·log₂(8) equals 4·3 B 4 because 4·log₂(8) equals 4·1 C 6 because 4·log₂(8) equals 4·1,5 D 8 because 4·log₂(8) equals 4·2

11 Using the change of base formula with base 2, what is log₄(8)?

A 1.5 because log₂(8)/log₂(4) equals 3/2 B 0.5 because log₂(8)/log₂(4) equals 1/2 C 2.5 because log₂(8)/log₂(4) equals 5/2 D 3 because log₂(8)/log₂(4) equals 3/1

12 Solve the logarithmic equation log₂(x) = 4 in the set of real numbers.

A x = 2 because 2⁴ equals 2 B x = 8 because 2⁴ equals 8 C x = 4 because 2⁴ equals 4 D x = 16 because 2⁴ equals 16

13 Solve log₃(x + 2) = 2 and respect the domain of the logarithm.

A x = 1 because x + 2 must equal 3² B x = 5 because x + 2 must equal 3² C x = 7 because x + 2 must equal 3² D x = 9 because x + 2 must equal 3²

14 What is the value of log₂(1/8) expressed as a power of 2?

A 2 because 1/8 equals 2² B 3 because 1/8 equals 2³ C −2 because 1/8 equals 2⁻² D −3 because 1/8 equals 2⁻³

15 Which statement correctly describes the behavior of the logarithmic function f(x) = log_a(x)?

A It is decreasing only when a equals exactly 1 B It is increasing for a > 1 and decreasing for 0 < a < 1 C It is increasing for 0 < a < 1 and decreasing for a > 1 D It is constant for all admissible positive bases a

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