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logarithm

/ˈlɔːɡəˌrɪðəm//ˈlɒɡərɪðəm/·noun
a mathematical operation that measures a quantity by its order of magnitude — in powers of ten
Fig. 1 — I climb a ladder whose rungs are spaced exactly evenly, and at first that looks reassuring.
01Definition

A logarithm answers one question: how many times must I multiply the base by itself to reach this number? The base-10 logarithm of 1000 is 3, because 10 × 10 × 10 = 1000. Coined in 1614 by John Napier from the Greek logos ('ratio') and arithmos ('number'), it turns multiplication into addition and compresses enormous ranges into a readable scale. That is why earthquakes, sound, and acidity are read on logarithmic scales, where each equal step is a tenfold jump in the real quantity — a hidden surge that the even spacing politely conceals. It rhymes with algorithm but shares none of its meaning.

02In use
  • iBecause the decibel is a logarithm, every statistic that looks like a small rise can hide a tenfold jump in real energy.
  • iiPlotting the figures on a logarithmic axis tamed a curve that would otherwise surge off the top of the page.
  • iiiStudents often confuse the logarithm with the algorithm, though one measures scale and the other lists steps.
03Collocations
  • the logarithm of
  • natural logarithm
  • base-10 logarithm
  • take the logarithm
  • a logarithmic scale

Family logarithmic (adjective) · logarithmically (adverb)

04Relations

=log, exponent, logarithmic function

05Compared
PairThe difference
logarithm vs algorithman algorithm is something you DO (steps); a logarithm is something you CALCULATE (scale).
06TOEFL & IELTS

Logarithm and its adjective logarithmic turn up in academic Reading and Listening, especially in science and data passages ('a logarithmic scale', 'grows logarithmically'). The exam trap is purely visual and aural: algorithm is a step-by-step procedure, logarithm is a mathematical operation on scale — the same letters jumbled, opposite fields. Stress falls on the first syllable, LOG-a-rithm.

07Asked
Why do earthquakes and decibels use logarithmic scales?
Because the real quantities span ranges too huge to chart, a logarithmic scale compresses them by turning multiplication into equal spacing. One step of earthquake magnitude means about ten times the ground shaking and roughly thirty-two times the released energy; a rise of ten decibels is ten times the sound power. The ladder in the scene above is honest about the trick: the rungs stay even while the numbers beside them explode.
Is logarithm related to algorithm?
Not at all, despite sharing nearly every letter — a logarithm measures scale while an algorithm lists steps, and the anagram trap has its own page, algorithm vs logarithm, in the 'Compared' list above.
What is the difference between log and ln?
The base. Plain log usually means the base-10 (common) logarithm — how many tens multiply to reach the number — while ln is the natural logarithm, built on the constant e ≈ 2.718, which appears wherever growth feeds on itself. The abbreviation is usually traced to the Latin logarithmus naturalis. Calculators keep separate buttons; science leans on ln, engineering and measurement scales on log.
Is a logarithm just the inverse of an exponent?
Yes — they ask opposite questions about the same equation. The exponent starts from the recipe and asks what it produces: ten to the third gives what? The logarithm starts from the result and asks for the recipe: a thousand needs how many tens? Because they undo each other, taking the log of a power hands back the exponent — which is precisely what lets multiplication be done by addition.
Who invented logarithms, and why?
John Napier, the Scottish mathematician who published them in 1614 after spending some twenty years computing his tables by hand. The point was labour-saving: astronomers drowning in many-digit multiplication could add logarithms instead. The invention held that job for three and a half centuries — printed log tables and the slide rule ran science and engineering until pocket calculators retired them in the 1970s.