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Mathematics · Class 9 · Chapter 2: Logarithms

8 min

Key points

Logarithm is the inverse operation of exponentiation — if aˣ = N, then log_a N = x
Common logarithm has base 10 (log N) and natural logarithm has base e (ln N)
Product rule: log_a (MN) = log_a M + log_a N
Quotient rule: log_a (M/N) = log_a M - log_a N
Power rule: log_a (Mⁿ) = n log_a M
Change of base formula: log_a N = (log_b N)/(log_b a)
log_a 1 = 0 and log_a a = 1 for any base a > 0, a ≠ 1
Antilogarithm is the inverse of logarithm — if log N = x, then antilog x = N
Logarithmic equations can be solved by converting to exponential form

Definitions

Logarithm: The logarithm of a positive number N to a positive base a (a ≠ 1) is the exponent to which the base a must be raised to get the number N. If aˣ = N, then log_a N = x.
Common Logarithm: A logarithm with base 10 is called common logarithm. It is written as log N instead of log₁₀ N.
Natural Logarithm: A logarithm with base e (where e = 2.718...) is called natural logarithm. It is written as ln N or log_e N.
Antilogarithm: If log N = x, then N is called the antilogarithm of x. We write N = antilog x or N = 10ˣ.

Worked example

Define logarithm. If log₃ x = 4, find the value of x. (4 marks)

Definition: The logarithm of a positive number to a given base is the power to which the base must be raised to get that number. Given: log₃ x = 4. By definition of logarithm: x = 3⁴. Therefore: x = 3 × 3 × 3 × 3 = 81

Common mistakes

Students confuse log_a b with log_b a — remember the base is written as subscript
Forgetting domain restrictions — logarithm is only defined for positive numbers
Mixing up product and quotient rules — log(MN) = log M + log N, not log M × log N
Not converting to exponential form when solving logarithmic equations
Calculating antilog incorrectly — if log N = 2, then N = 10², not 2 × 10

Quick recap

Logarithm is the power to which a base must be raised to get a given number. Common logarithm uses base 10, natural logarithm uses base e. Always convert log equations to exponential form for solving. Remember the fundamental relationship: if logₐ x = n, then aⁿ = x.

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