Go to options -> interface -> console -> uncheck "show info messages" "show event messages" "show status messages" and "show status messages of others" if you hover over the explanation mark on the side it will tell you what these options do if enabled. this will make sure in your server log you don't see your or other peoples damage, condition, raid messages, green messages like you using runes etc. The ones you can leave checked would be the time stamps and levels so you can see peoples levels and the time in their messages.
This isn't to do with the Server Log but just a suggestion: You can always open up the loot chat to see what you loot from bosses which you can do by opening up the loot channel then right clicking on the tab "show in read only tab" shown in picture below. However, keep in mind when you right click the loot channel make sure you don't select "show server messages" if you do simply right click again and "hide server messages" to avoid anything else popping up in the chat.
You can also open up a read-only tab by dragging and dropping a chat on where the pink highlight is in picture below which is the top right of your chat box. Enjoy!