First off, I believe that there are three basic types of logs: duty, personal and story. Duty Logs are generally recaps what you did during the sim. In short, they are a synopsis of the tasks you and the others in your department performed. Duty logs are important not only because they help you get promoted, but also because they help your Commanding Officer see what everyone was doing during the sim. It is very hard to know exactly what's going on in the sim at every single moment, because there are often "subplots" involved in sims (ie - Engineering may be working on recalibrating the warp core, Sickbay may be treating an outbreak of poison ivy, and the CO & XO may be hosting an Ambassador). Going over the duty logs is an excellent way for the Captain to steer the sim in a certain direction, based on what the other crew members have been doing. Personal Logs generally are logs which include your character's personal feelings about the current plot. You may wish to note that you think your department head is going through some form of crisis, or that your Captain has a large growth coming off of his left toenail. Personal Logs basically allow you to explore your creativity and provide some "behind the scenes" information on what your character was feeling, thinking, etc. Story Logs generally have nothing to do with the current plot. Story logs are used for character development and can have their own plot all by themselves. For example, you may wish to construct a story log based on the death of your character's sister. You could write the entire thing out from the time you received the transmission, to the time you went to Earth for the funeral, to the time that you had to consult with the ship's counselor for support. The nice thing about story logs is that they do not fall within the current setting of the sim, so you could have your story occur 10 years in the past, or 3 days in the future -- just make sure that your story doesn't create a very large impact on the actual sim. (ie - the Captain slips on the XOs banana peel and dies).
I would recommend being respectful when creating your logs. If you're going to make another member of the crew feel something or do something, check with them to make sure it's OK first -- you can always create a Non-Personnel Character (NPC) or two to help your logs along. Personally, I wouldn't like it very much if somebody was writing about having a romantic involvement with me, as it would go against my character's history - and I may already have my eye on the Lieutenant in Stellar Cartography.
While most simmers write logs, not many people understand why they are important. Duty logs are important because they actually help the sim get built up, and provide a record for the sim's history. As a Commanding Officer, I always review all of the logs from the previous week before I send out the briefing to ensure that I don't completely remove a subplot. Personal Logs & Story Logs, however, are important depending on how serious of a simmer you are. Personal & Story logs provide wonderful ways for you to let everyone know about the history of your character as well as events which may happen in his or her life. Personally, I spent a lot of time creating my character, Tr'Vek, and I like to continue working on him through story logs. As I'm sure you know, many times on Star Trek, the episode may revolve around a specific person -- such as Counselor Troi's pesky mother. Unfortunately, it is very difficult to sim these situations out, because the rest of the crew would be very bored! Story logs are a wonderful way for you to do this -- and the best part is -- it's YOUR creation. You control what happens and who does what.
I hope that this helps all of you in your venture to become FSF officers. I have found that any of the officers on my sim (Cassini Station) that have gone through instruction at the FSF Academy are skilled and knowledgeable, and I believe that you will all make your future Commanders very proud. I look forward to reading the logs you create, and I welcome you to send them to me for my opinion.
Although some sims in the FSF do not require that you post your log to a message board, I would highly recommend that you do. I have been a member of the simming world for many years, and I find it fun to go to the old message boards and read some of my logs from the past. It very well may amaze you one day to see how far you've come -- I know it has amazed me very much!