Logarithm Rules and Computation

Understanding the properties of logarithms and how to apply them in calculations

log

Logarithm Basics

A logarithm is the inverse operation to exponentiation. The logarithm of a number is the exponent to which the base must be raised to produce that number.

log_b(a) = c ⇔ b^c = a

Where:

• b is the base (b > 0, b ≠ 1)
• a is the argument (a > 0)

Common Logarithms and Natural Logarithms

Common logarithm: Base 10, written as log(x)

Natural logarithm: Base e (≈ 2.718), written as ln(x)

x

y

Logarithm Rules

Product Rule

log_b(M × N) = log_b(M) + log_b(N)

Example: log₂(8 × 4) = log₂(8) + log₂(4) = 3 + 2 = 5

Quotient Rule

log_b(M ÷ N) = log_b(M) - log_b(N)

Example: log₃(81 ÷ 9) = log₃(81) - log₃(9) = 4 - 2 = 2

Power Rule

log_b(M^p) = p × log_b(M)

Example: log₅(25³) = 3 × log₅(25) = 3 × 2 = 6

Change of Base Formula

log_b(a) = log_c(a) ÷ log_c(b)

Example: log₂(9) = log₁₀(9) ÷ log₁₀(2) ≈ 0.954 ÷ 0.301 ≈ 3.17

Important Properties

• log_b(1) = 0 for any base b (since b⁰ = 1)
• log_b(b) = 1 for any base b (since b¹ = b)
• log_b(b^c) = c by the definition of logarithms
• b^log_b(a) = a by the definition of logarithms
• Logarithms are only defined for positive arguments
• Logarithmic functions are the inverses of exponential functions

Try It Yourself

Calculate log₁₀ of a number:

log₁₀( )

Result: log₁₀(100) = 2

Change of base: Calculate log_b(a)

log( )

Result: log₂(8) = 3

Math Learning Resources | Logarithm Rules and Computation