Where did the word "logarithm" come from?

Question

Where did the word logarithm come from? Any relation to the word algorithm?

Answer 1

There is no relation between the two words.

logarithm: 1610s, Mod.L. logarithmus, coined by Scottish mathematician John Napier (1550-1617), lit. "ratio-number," from Gk. logos "proportion, ratio, word"

algorithm: was derived from the name of 8th century Persian mathematcian al-Kwarizmi.

Note: I think it's unusual for a term to derive from a person's name, especially in mathematics. I know words like "bowdlerize" (meaning to edit by removing offensive material) from Thomas Bowdler, or a "spoonerism" (a phrase constructed by exchanging syllables between words, eg "Swell foop") named after William Spooner, but in math I believe it's quite rare. The now-standard lowercase "abelian" is perhaps another example.

Answer 2

"There is no relation between the two words." feels wrong. The spelling of algorithm was influenced by the Greek árithmos, which is the second compound in logarithm.

I quote from Word Origins (2005 2e) by John Ayto. p 16 Left column.

algorithm [13]

Algorithm comes from the name of a Persian mathematician, in full Abu Ja far Mohammed ibn-Musa al-Khwarizmi (c. 780–c. 850), who lived and taught in Baghdad and whose works in translation introduced Arabic numerals to the West. The last part of his name means literally ‘man from Khwarizm’, a town on the borders of Turkmenistan, now called Khiva.
      The Arabic system of numeration and calculation, based on 10, of which he was the chief exponent, became known in Arabic by his name – al-khwarizmi. This was borrowed into medieval Latin as algorismus (with the Arabic -izmi transformed into the Latin suffix -ismus ‘-ism’). In Old French algorismus became augorime, which was the basis of the earliest English form of the word, augrim. From the 14th century onwards, Latin influence gradually led to the adoption of the spelling algorism in English. This remains the standard form of the word when referring to the Arabic number system; but in the late 17th century an alternative version, algorithm, arose owing to association with Greek árithmos ‘number’ (source of arithmetic [13]), and this became established from the 1930s onwards as the term for a step-by- step mathematical procedure, as used in computing.
      Algol, the name of a computer programming language, was coined in the late 1950s from ‘algorithmic language’.

Op cit. p 314 Right column.

logarithm [17]

Greek lógos had a remarkably wide spread of meanings, ranging from ‘speech, saying’ to ‘reason, reckoning, calculation’, and ‘ratio’. The more ‘verbal’ end of its spectrum has given English the suffixes -logue and -logy (as in dialogue, tautology, etc), while the ‘reasoning’ component has contributed logic [14] (from the Greek derivative logiké), logistic [17] (from the Greek derivative logistikós ‘of calculation’), and logarithm, coined in the early 17th century by the English mathematician John Napier from Greek logós ‘ratio’ and arithmós ‘number’ (source of English arithmetic [13]).