Logarithms: The Inverse of Exponentiation
A logarithm answers the question: "To what power must I raise the base to get this number?" If 2³ = 8, then log₂(8) = 3. Logarithms are essential in science (pH, decibels, Richter scale), finance (compound growth), and computing (algorithm complexity).
Definition
log_b(x) = y ↔ b^y = x
log₁₀(1000) = 3 (because 10³ = 1000)
log₂(32) = 5 (because 2⁵ = 32)
ln(e) = 1 (because e¹ = e)The Three Common Bases
- log₁₀ (common log): Used in chemistry (pH), acoustics (dB), seismology
- ln (natural log, base e≈2.718): Used in calculus, finance, statistics
- log₂ (binary log): Used in computing, information theory
Key Laws of Logarithms
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) = ln(x) ÷ ln(b)Worked Examples
- log₁₀(500) = log₁₀(1000) − log₁₀(2) ≈ 3 − 0.301 = 2.699
- ln(100) = ln(e^4.605) ≈ 4.605
- log₂(64) = 6 (because 2⁶ = 64)
Calculate any logarithm: Free Logarithm Calculator
The Three Common Logarithms
- log₁₀ (common log, written log): Base 10, used in chemistry (pH), acoustics (decibels), earthquake magnitude (Richter scale), and signal strength.
- ln (natural log): Base e ≈ 2.71828, used in calculus, continuous growth/decay models, statistics, and engineering. Most fundamental mathematically.
- log₂ (binary log): Base 2, used in computer science (binary, entropy, information theory). The number of bits needed to represent n values = log₂(n).
Key Properties and Rules
Logarithm rules that are frequently useful: Product rule: log(AB) = log(A) + log(B). Quotient rule: log(A/B) = log(A) - log(B). Power rule: log(Aⁿ) = n·log(A). Change of base: log_b(x) = ln(x)/ln(b). These rules allow multiplication and division to be performed as addition and subtraction — the basis of how slide rules and early logarithm tables worked. Logarithms compress large ranges of numbers, which is why the Richter scale (earthquakes), pH scale (acidity), and decibel scale (sound) all use logarithmic notation.
Frequently Asked Questions
What is log(0) or log of a negative number?
log(0) is undefined — there is no power of any base that equals zero. log of a negative number is also undefined in real numbers (though complex logarithms do exist). This reflects the fact that exponential functions (bˣ) are always positive for positive b, so their inverse — the logarithm — only accepts positive inputs.
How is pH calculated from hydrogen ion concentration?
pH = -log₁₀[H⁺], where [H⁺] is the molar concentration of hydrogen ions. Pure water has [H⁺] = 10⁻⁷ mol/L, so pH = -log(10⁻⁷) = 7. An acid with [H⁺] = 10⁻³ mol/L has pH = 3. Each pH unit represents a tenfold change in acidity — pH 3 is 10,000 times more acidic than pH 7.
Why is e the natural base for logarithms?
The number e ≈ 2.71828 arises naturally when dealing with continuous growth. The derivative of eˣ is eˣ (unique among all exponential functions), making ln(x) the simplest logarithm to differentiate and integrate. Any exponential growth or decay process — population growth, radioactive decay, compound interest — is most cleanly expressed using e and ln.