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logarithm

/ˈlɔːɡəˌrɪðəm//ˈlɒɡərɪðəm/·noun
I climb a ladder whose rungs are spaced exactly evenly, and at first that looks reassuring. But the numbers painted beside them read one, ten, a hundred, a thousand — each rung worth ten of the one below. A glowing level rises one even step, and the value behind it has already multiplied tenfold; another even step, and it multiplies again. By the top the quantity is colossal, yet I only ever took four calm, equal strides to get there. The ruler stays even on purpose: it is the size that explodes, not the spacing.
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Definition

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.

Examples

  • Because the decibel is a logarithm, every statistic that looks like a small rise can hide a tenfold jump in real energy.
  • Plotting the figures on a logarithmic axis tamed a curve that would otherwise surge off the top of the page.
  • Students often confuse the logarithm with the algorithm, though one measures scale and the other lists steps.

Collocations

the logarithm of·natural logarithm·base-10 logarithm·take the logarithm·a logarithmic scale

Synonyms

log·exponent·logarithmic function

See also

Word family

logarithmic (adjective)·logarithmically (adverb)

In TOEFL & 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.