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