What Is a Logarithm?
A logarithm answers the question: to what power must we raise the base to get a given number? log_b(x) = y means b^y = x.
log₁₀(1000) = 3 because 10³ = 1000. ln(e²) = 2 because e² = e². log₂(32) = 5 because 2⁵ = 32.
Logarithms were invented by John Napier in 1614 to simplify complex multiplication and division — before calculators, multiplying large numbers meant looking up logarithm tables, adding the logarithms, and looking up the antilogarithm. This reduced hours of arithmetic to minutes.
Logarithmic Scales in the Real World
Logarithmic scales compress enormous ranges into human-readable numbers. The Richter scale measures earthquake magnitude: each whole number represents a 10× increase in amplitude and about 31.6× more energy. An earthquake of magnitude 6.0 releases 31.6 times more energy than a 5.0.
Decibels (dB) measure sound intensity logarithmically: 60 dB (normal conversation) is 10 times louder than 50 dB (quiet office), not twice as loud. 120 dB (rock concert) is 10^6 = 1,000,000 times more intense than 60 dB.
pH chemistry: each pH unit represents a 10× change in hydrogen ion concentration. pH 4 is 10 times more acidic than pH 5. pH 1 (stomach acid) is 10^6 = 1,000,000 times more acidic than pH 7 (neutral water).