A logarithm is the inverse of a power (exponentiation). If 10² = 100, then log₁₀(100) = 2. The logarithm asks: 'To what power must I raise the base to get this result?' Log base 10 of 1,000 is 3 because 10³ = 1,000. Log base 2 of 8 is 3 because 2³ = 8. Log base e (the natural logarithm, written ln) of e² is 2.
The most common bases are 10 (common logarithm, log), e ≈ 2.71828 (natural logarithm, ln), and 2 (binary logarithm, log₂). Calculators typically have buttons for log (base 10) and ln (base e). For other bases, use the change of base formula: logₐ(x) = log(x) / log(a) or ln(x) / ln(a).
Why logarithmic scales are everywhere
Logarithms are used to express quantities that span many orders of magnitude on a manageable scale. The Richter scale for earthquakes is logarithmic: a magnitude 7.0 earthquake releases 31.6 times more energy than a 6.0 earthquake (because 10^1.5 ≈ 31.6, where 1.5 is the energy exponent per Richter unit). A magnitude 9.0 earthquake releases 10^4.5 ≈ 31,623 times more energy than a 6.0 — logarithmic scaling makes these extreme differences readable.
Sound intensity (decibels), pH (acidity), stellar magnitude, and information theory (bits as log₂) all use logarithmic scales for the same reason. pH = −log₁₀([H⁺]), so a pH of 4 has 10 times more hydrogen ions than a pH of 5, and 100 times more than pH 6. Decibels: 0 dB is the threshold of hearing, 60 dB (a normal conversation) is 10^6 times more intense, and 120 dB (a jet engine) is 10^12 times more intense.
Key properties that make logarithms useful in calculation
Logarithms turn multiplication into addition (log(a × b) = log(a) + log(b)) and exponentiation into multiplication (log(aⁿ) = n × log(a)). Before calculators existed, this is why slide rules and logarithm tables were used for complex calculations: multiplying large numbers becomes adding their logarithms, then looking up the antilogarithm. This was how NASA engineers calculated rocket trajectories by hand.
In finance and economics, logarithmic returns (also called log returns or continuously compounded returns) are preferred over simple percentage returns because they're additive over time. If a stock gains log(1.10) in year 1 and log(0.90) in year 2, the total two-year log return is simply their sum. Simple percentage returns (10% and −10%) compound non-symmetrically and don't add, making multi-period analysis more complicated.