Logarithms Demystified: log, ln & Change of Base Explained

1. What is a Logarithm?

A logarithm answers the question: "To what power must the base be raised to produce this number?"

Logarithmic notation is simply the inverse of exponential notation:

log_b(y) = x ⟺ bˣ = y

For example, because 10² = 100, we write log₁₀(100) = 2.

2. Common Log (log) vs Natural Log (ln)

• Common Log (log): Implies base 10. Frequently used in acoustics (decibels), chemistry (pH scale), and earthquake measurement (Richter scale).
• Natural Log (ln): Uses Euler's number e ≈ 2.718281828 as its base. Essential in continuous compound interest, radioactive decay, population growth, and calculus derivatives.

3. Fundamental Logarithmic Laws

• Product Rule: log(M × N) = log(M) + log(N)
• Quotient Rule: log(M / N) = log(M) − log(N)
• Power Rule: log(Mᵏ) = k × log(M)
• Identity: log_b(b) = 1 and log_b(1) = 0

4. Change of Base Formula

If your calculator only has 'log' (base 10) and 'ln' (base e), you can evaluate any arbitrary base using the change-of-base formula:

log_b(x) = ln(x) / ln(b) = log(x) / log(b)

Example: To calculate log₂(32): ln(32) / ln(2) = 3.4657 / 0.6931 = 5 (since 2⁵ = 32).