Where did the word "logarithm" come from? Where did the word logarithm come from? Any relation to the word algorithm? 
 A: 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.
A: "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]).

