A ships writer is in charge of reports; gear, personnel and to some extent pay, though that was formally handled by the ship's paymaster. Nearly every western naval ship post 1600 had one or someone who shared that responsibility.

In a cursory search, I cannot find proof that such a role involved official keeping of the Ship's Log, but neither have I found contrary evidence.

Their job would be to log and collate information about the ship, one site suggests that the ship 'deck log' would contain the following;

Absentees, Accidents [material], Accidents/Injuries [personnel], Actions [combat], Appearances of Sea/Atmosphere/Unusual Objects, Arrests/Suspensions, Arrival/Departure of Commanding Officer, Bearings [navigational], Cable/Anchor Chain Strain, Collisions/Groundings, Courts-Martial/Captain's Masts, Deaths, Honors/Ceremonies/Visits Incidents at Sea, Inspections, Meteorological Phenomena, Movement Orders, Movements [getting underway; course, speed changes; mooring, anchoring], Passengers, Prisoners [crew members captured by hostile forces], Propulsion Plant Status changes, Ship's Behavior [under different weather/sea conditions], Sightings [other ships; landfall; dangers to navigation], Soundings [depth of water], Speed Changes Tactical Formation, Time of Evolutions/Exercises/Other Services Performed.

(Source: http://www.history.navy.mil/faqs/faq73-1.htm)

A column in the 'Geraldton Guardian', a West Australian newspaper published in 19th November 1918 opines that the Ship's Writer 'first and foremost virtue is an exhaustive knowledge of the King's Regulations: he is the oracle of the lower deck on all matters relating to pay and promotion'. It also states that the Ship's Writer would serve as an assistant to the Paymaster and also act as a clerk or notary when the Captain of the vessel sat in judgement upon a member of the crew in a formal court. In battle his job was to assistant carrying wounded men or ammunition (and in theory, attachment to any sort of 'working party') after which he would 'retire to the office for a little light recreation with the ship's ledger.', clearly referring to the chaos of after battle reports; noting structural damage, loss of equipment, spent shot and powder and current levels, any flooding, not to mention the casualties, deaths and injuries.

In any case he definitely 'prepares the numerous returns relating to the ship's gear and personnel and he conducts the ship's correspondence.'

(Source: http://trove.nla.gov.au/ndp/del/article/66506716)

As a point of interest, the New York Times in 1893 published a column titled 'Fall of the Ship's Writer' bemoaning the loss of status and pay of the position, stating that 'to the crew the First Lieutenant is regarded as the power behind the throne, but in many cases his Writer was a power that was greater than the throne itself.'

(Source: http://query.nytimes.com/mem/archive-free/pdf?res=F7061EFB345B1A738DDDA00994DB405B8385F0D3)

Having tracked an elusive citation to 'The End of Barbary Terror : America's 1815 War against the Pirates of North' by Frederick C. Leiner, who cites the USS Macedonian's log for 1815-1816, it seems possible that if a log survived for the time period of 1864-5, it may be held where this record is; in the National Archives (the United States I believe)

'Deck logs of commissioned U.S. Navy ships from the earliest times through 1940 are in the Old Military and Civil Branch, National Archives and Records Administration, 700 Pennsylvania Avenue NW, Washington DC 20408 [Telephone (202) 501-5385,web-site http://www.archives.gov/dc-metro/washington/]'

Apologies for the messy answer, my specialty is not Naval History. However it is 'frantic research past midnight.' I hope it helps!