Log Calculator

The logarithm is the inverse of exponentiation — it answers "what power must we raise the base to in order to get this number?" This calculator handles log base 10, natural log (ln), log base 2, and any custom base.

How to Use This Calculator

1. Enter the number (argument) — must be positive.
2. Select the base: log₁₀ (common log), ln (natural log, base e), log₂ (binary log), or custom base.
3. Click Calculate to see the logarithm value and the equivalent exponential equation.

Logarithm Definition

log_b(x) = y means bʸ = x

Example: log₁₀(1000) = 3 because 10³ = 1000.

Common Logarithm Types

• log (base 10): Common log. log(100) = 2, log(1000) = 3. Used in pH, decibels, Richter scale.
• ln (base e): Natural log. ln(e) = 1, ln(1) = 0. Used in calculus, exponential growth, compound interest.
• log₂ (base 2): Binary log. log₂(8) = 3, log₂(1024) = 10. Used in computer science, information theory.

Key Logarithm Rules

• Product: log(ab) = log(a) + log(b)
• Quotient: log(a/b) = log(a) − log(b)
• Power: log(aⁿ) = n × log(a)
• Change of base: log_b(x) = log(x) / log(b) = ln(x) / ln(b)

Example Calculations

• log₁₀(10,000) = 4 (because 10⁴ = 10,000)
• ln(e²) = 2
• log₂(64) = 6 (because 2⁶ = 64)
• log₅(125) = log(125)/log(5) = 2.097/0.699 = 3

Common Mistakes to Avoid

• log(a + b) ≠ log(a) + log(b) — This is one of the most common log errors. Only log(a × b) = log(a) + log(b).
• Taking log of a negative number — Log is undefined for zero and negative numbers in real mathematics.
• Confusing log and ln on calculators — The "log" button is base 10; the "ln" button is natural log (base e). They give very different results.

Frequently Asked Questions

What is the natural log used for?

The natural log appears in calculus (derivatives of exponential functions), continuous compound interest (A = Peʳᵗ), population growth models, thermodynamics, and probability distributions.

How is log related to exponents?

Logs and exponents are inverse operations: if 10³ = 1000, then log₁₀(1000) = 3. Understanding this inverse relationship is the key to using logarithms effectively.

What is pH in terms of logarithms?

pH = −log₁₀[H⁺], where [H⁺] is hydrogen ion concentration. A pH of 7 means [H⁺] = 10⁻⁷ mol/L. Each pH unit represents a 10× change in acidity.

Conclusion

Logarithms are essential in science, finance, and computer science — anywhere exponential growth or measurement scales appear. Use this calculator for any logarithm computation, and use the rules above to simplify complex log expressions by hand.

Related: Exponent Calculator | Scientific Calculator | Half Life Calculator