Logarithms: Concept, Laws & Principles of Calculation

1. Definition and Equivalence with Exponential Form

If a is a positive real number other than 1 (a > 0, a ≠ 1), and aˣ = N, then the exponent x is called the logarithm of N to the base a:

aˣ = N ⟺ x = log_a(N)

Equivalence between Exponential Form and Logarithmic Form

Example 1

Exponential: 2⁵ = 32
Logarithmic: log₂(32) = 5

Example 2

Exponential: 10³ = 1,000
Logarithmic: log₁₀(1000) = 3

Example 3

Exponential: 7⁰ = 1
Logarithmic: log₇(1) = 0

The Core Operational Laws of Logarithms

Product Law log_a(m × n) = log_a(m) + log_a(n):

The logarithm of a product is equal to the sum of the logarithms of the individual numbers.

Quotient Law log_a(m ÷ n) = log_a(m) - log_a(n):

The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.

Power Law log_a(mᵖ) = p × log_a(m):

The logarithm of a number raised to a power equals the power multiplied by the logarithm of that number.

Change of Base log_b(m) = log_a(m) ÷ log_a(b):

Enables conversion of logarithms from any arbitrary base b to standard base 10 for evaluation using logarithmic tables.

log_a(a) = 1 (The logarithm of any base to itself is always 1; e.g., log₁₀(10) = 1).
log_a(1) = 0 (The logarithm of 1 to any base is always 0; e.g., log₁₀(1) = 0).
Logarithms of negative numbers and zero do not exist in the real number system.
Common Logarithms: Logarithms using base 10 (denoted as log or log₁₀), used universally in commercial calculations.
Natural Logarithms: Logarithms using Euler’s constant e ≈ 2.718 (denoted as ln), used in continuous compounding and economic growth modeling.