The log reveals Michael's attempt to communicate with Walt.
Each of the Numbers printed on the log increase by 108 minutes (or seconds) eg. 41602064:41 + 1:48 (60+48 = 108) = 41602066:29 This therefore puts forward the possibility that each time the Numbers are entered, it is 'accepted'.
If 'accepted' refers to the time the Numbers were entered, then the entry would have to be every 108 minutes, precisely, which is not accurate given the various entry occurances within (or after) the available 4 minute entry period.
However, while the Numbers might refer to each 108 minute interval, 'accepted' might simply indicate completion of the entry and acceptance by the computer, irrespective of the exact time the entry was accomplished.
Also, when the Numbers are entered in the 4 minute period, the amount of time remaining in the 108 minute period could be added onto the next, without the inhabitants of the Hatch realizing (there being no other way of telling the time in the Hatch, they would never realize an extra few minutes added onto the time) If the 108 minutes therefore varied slightly, it is unlikely it is for any sort of scientific use.
It could also be that for the specific time period that this particular log is keeping track of, whoever was doing the button pushing may have actually waited until the last possible minute to hit 'execute.' This could explain why the count interval, though repeating, does not match up with a :00 start (i.e. at some other window of time, the button pusher did it at various times within the 4 minute window).
The log printout shows the hours and minutes that have passed since an event in the 28th century BC. For example: 41602050 hours would represent 4749.09 years. We don't know what (island) time period the print-out covers but it can reasonably be assumed to be between sometime not long after 1970 (the incident requiring the button to be pushed occurred "shortly after experiments began") and late 2004 (log printed out). This means that 0 on the log would represent a date sometime between 2775 BC and 2741 BC. An important person in that time period was Fu Hsi, legendary ruler of China (traditional dates: 2852 BC-2738 BC). According to tradition Fu Hsi had the arrangement of the trigrams of the Yi Jing revealed to him supernaturally. The trigrams form part of the Bagua which is used as the logo of the Dharma Initiative. The starting timestamp for the log could be the number of hours that have passed since the time the trigrams were revealed to Fu Hsi.
If the number to the right of the : is base 60 (like counting seconds and minutes), the number consistently increments by 1:48 each time.
If we convert 1:48 in base 60 to decimal, the result is 108 units.
We do not know whether the units represent hours, minutes, seconds, or something completely different.
The number after the : keeps repeating in the pattern 17 05 53 41 29 because incrementing by 1:48 in base 60 will always yield a repeating pattern of five number.
It appears that the pattern began at :17, :05, :53, :41, or :29, and not on :00 as most timers do.
The Numbers can be input 4 minutes before the timer reaches 0. Unless the Numbers are entered as the timer reaches 0 each time (which we know does not happen) then there would be a gradual shift from the :17, :05, :53, :41, :29, pattern.
Since the countdown time does not display seconds until the 4 minute mark, it is possible that the timer slows down (or stays on 108 for more than 1 minute) to catch up with any shift.
Nature of the Printout
The printout could have been modified or forged entirely and planted so Locke and the rest would find it, which would explain the exact (and unlikely) 108-minute interval between "accepted" messages.
The "shift" from 17, 05, 53, 41, and 29 to 16, 04, 52, 40, and 28 might be a clue that the printout is not genuine.