Log Calculator Calculators

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A logarithm (log) answers the question: to what power must the base be raised to produce a given number? If log₁₀(x) = y, then 10^y = x. Logarithms convert multiplicative relationships to additive ones, making them essential for handling the enormous ranges of values common in science — from pH (hydrogen ion concentration spanning 14 orders of magnitude) to sound intensity (decibels) to earthquake magnitude (Richter scale) to star brightness (magnitude) and optical density in spectrophotometry.

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Log₁₀ (Common Logarithm)

log₁₀(x) = y ↔ 10^y = x

Key values: log(1) = 0; log(10) = 1; log(100) = 2; log(0.001) = −3; log(500) ≈ 2.699.

Calculator: 'log' key. Excel: =LOG10(x) or =LOG(x,10).

Log Properties

  • log(a × b) = log(a) + log(b)
  • log(a/b) = log(a) − log(b)
  • log(a^n) = n × log(a)
  • log(10^x) = x

Change of Base Formula

log_b(x) = log₁₀(x) / log₁₀(b) = ln(x) / ln(b)

Convert any log to base 10 or natural log: log₂(8) = log(8)/log(2) = 0.903/0.301 = 3. In Excel: =LOG(x, base).

Scientific Applications

  • pH: pH = −log₁₀[H⁺]; each unit = 10-fold [H⁺] change
  • Absorbance: OD = log₁₀(I₀/I)
  • Decibels: dB = 10 × log₁₀(I/I₀)
  • Richter scale: Each unit = 10× amplitude, ~31.6× energy
  • qPCR standard curve: Ct vs. log₁₀(copy number)
  • Margalef index: D = (S−1)/ln(N) uses natural log

Glossary

Logarithm (log₁₀)
The power to which 10 must be raised to produce a given number: log₁₀(x) = y means 10^y = x; converts multiplicative relationships to additive; used in pH, OD, and qPCR calculations.
Change of Base Formula
log_b(x) = ln(x)/ln(b) = log₁₀(x)/log₁₀(b); converts logarithms to any base using natural or common log; used in Excel as =LOG(x,b).
Antilogarithm (Base 10)
The inverse of log₁₀: antilog(y) = 10^y; used to convert pH to [H⁺] (10^−pH), Ct values to copy numbers, and OD measurements back to transmittance.

Frequently Asked Questions

A logarithm answers 'to what power must the base be raised to get this number?' log₁₀(x) = y means 10^y = x. Examples: log₁₀(1000) = 3 (10³ = 1000); log₁₀(0.01) = −2 (10⁻² = 0.01). Calculator: use the 'log' key for base-10. In Excel: =LOG10(x). For the natural logarithm (base e): use the 'ln' key or =LN(x). For any other base: log_b(x) = ln(x)/ln(b) or LOG(x,b) in Excel.

Key properties: (1) log(a×b) = log(a) + log(b) — log converts multiplication to addition. (2) log(a/b) = log(a) − log(b) — log converts division to subtraction. (3) log(a^n) = n×log(a) — exponents become coefficients. (4) log(1) = 0 for any base. (5) log_b(b) = 1 always. These properties make logarithms powerful for simplifying calculations with very large or very small numbers and for linearizing exponential relationships in graphs.

pH = −log₁₀[H⁺]. The negative log converts the tiny H⁺ concentrations (10⁻³ to 10⁻¹⁴ M) to a convenient 0–14 scale. Each pH unit represents a 10-fold change in [H⁺]: pH 3 has 10× more H⁺ than pH 4. To find [H⁺] from pH: [H⁺] = 10^(−pH). Example: pH 7.4 → [H⁺] = 10^(−7.4) = 3.98 × 10⁻⁸ M. The pKa of weak acids similarly uses −log(Ka) and is used in Henderson-Hasselbalch buffer calculations.

Change of base formula: log_b(x) = log₁₀(x)/log₁₀(b) = ln(x)/ln(b). Needed when working with bases other than 10 or e. Examples: log₂(64) = log(64)/log(2) = 1.806/0.301 = 6. log₃(81) = log(81)/log(3) = 1.908/0.477 = 4. Common uses in biology: log₂ fold change in RNA-seq (log₂(treated/control)); log₂(colony count) in some microbiology applications; any situation where a non-standard base simplifies the biology.