Server log analysis using machine learning
I don't think you necessarily need to convert the individual log entries into vectors for use in an algorithm.  I would guess that what you are interested in is a sequence of log entries, which represent a series of events, ordered in time, which together make up a series of 'cases'.  Here the relationship between a series of collected log entries is important.

If this is the case then you might want to consider using Process Mining techniques.  This allows you to build models of your process (the use of your application) and determine patterns of process steps, along with errors and rework steps.

There is a good introduction course on Coursera, here.  There are even some developed, commercial packages like 'disco' to help you with the analysis and visualisation