Logarithm Calculator Calculators

0 calculators tagged with “Logarithm Calculator”

A logarithm answers the question: to what power must a base be raised to produce a given number? Written log_b(x) = y means b^y = x. Logarithms are the inverse of exponential functions and appear throughout science — pH is a negative log of hydrogen ion concentration, decibels use a log scale for sound intensity, and the Richter scale uses logs to express earthquake magnitude. Mastering logarithm rules and being able to change between bases is a core quantitative skill in chemistry, biology, and data analysis.

All Calculators

No calculators found for this topic.

Logarithm Definitions

log_b(x) = y ↔ b^y = x

  • Common logarithm (log₁₀ or log): Base 10. log₁₀(1000) = 3 because 10³ = 1000
  • Natural logarithm (ln): Base e ≈ 2.71828. ln(e²) = 2
  • Binary logarithm (log₂): Base 2. Used in information theory and molecular biology (e.g., fold change). log₂(8) = 3

Key Logarithm Rules

  • Product rule: log(xy) = log x + log y
  • Quotient rule: log(x/y) = log x − log y
  • Power rule: log(xⁿ) = n log x
  • Change of base: log_b(x) = log(x) / log(b) = ln(x) / ln(b)
  • log(1) = 0; log_b(b) = 1 for any base b

Worked Examples

log₁₀(500) = log₁₀(5 × 100) = log₁₀(5) + log₁₀(100) = 0.699 + 2 = 2.699

ln(50) = ln(5 × 10) = ln(5) + ln(10) = 1.609 + 2.303 = 3.912

log₂(32) = log₂(2⁵) = 5

Applications in Science

  • pH: pH = −log₁₀[H⁺]
  • Henderson-Hasselbalch: pH = pKa + log([A⁻]/[HA])
  • Beer-Lambert (absorbance): A = −log₁₀(T) = εcl
  • Decibel scale: dB = 10 × log₁₀(I/I₀)
  • Michaelis-Menten linearization: Lineweaver-Burk uses reciprocal plots
  • qPCR fold change: log₂(FC) from ΔΔCt method
  • Exponential growth: ln(N/N₀) = rt

ln vs log₁₀ Conversion

ln(x) = 2.303 × log₁₀(x)
log₁₀(x) = ln(x) / 2.303 = 0.4343 × ln(x)

Glossary

Logarithm
The inverse of exponentiation: log_b(x) = y means b^y = x. Answers 'to what power must b be raised to equal x?' Common bases: 10 (log), e (ln), 2 (log₂).
Change of Base Formula
log_b(x) = log(x)/log(b) = ln(x)/ln(b). Converts logarithms from one base to another. Useful when a calculator only has log₁₀ and ln functions.
Natural Logarithm (ln)
The logarithm with base e (≈2.71828). Denoted ln(x). Preferred in calculus-derived equations because d/dx[ln(x)] = 1/x. Related to log₁₀ by: ln(x) = 2.303 × log₁₀(x).

Frequently Asked Questions

log (or log₁₀) is the common logarithm — base 10. It answers: 10 to what power gives x? Used in pH, decibels, and Richter scale. ln is the natural logarithm — base e (≈2.71828). Used in calculus, growth/decay equations, and thermodynamics. log₂ is the binary logarithm — base 2. Used in information theory and molecular biology (gene expression fold change). All are related by the change of base formula: log_b(x) = ln(x)/ln(b).

Product: log(xy) = log x + log y. Quotient: log(x/y) = log x − log y. Power: log(xⁿ) = n log x. Change of base: log_b(x) = log(x)/log(b). Special values: log(1) = 0 for any base; log_b(b) = 1. These rules apply to any base (10, e, 2) and are used to simplify complex logarithmic expressions.

Use logarithm properties and known values. Common log₁₀ values to memorize: log(2) ≈ 0.301; log(3) ≈ 0.477; log(5) = 1 − log(2) ≈ 0.699; log(7) ≈ 0.845. For other values: log(6) = log(2) + log(3) ≈ 0.778; log(50) = log(5) + log(10) = 0.699 + 1 = 1.699. Scientific calculators give more precise values instantly.

ln appears in many biological equations: exponential population growth — ln(N/N₀) = rt; radioactive decay and half-life — t½ = ln(2)/k ≈ 0.693/k; the Michaelis-Menten linearization; Nernst equation for membrane potential; and allometric scaling equations. Natural logarithm is preferred in equations derived from calculus because the derivative of ln(x) is simply 1/x.