So for logging you can not add it into the time/date field but you can use the x-duration which is the seconds/milliseconds since wowza started - so you could use that field instead
When you start wowza in the logs one of the first lines logged is2020-01-23 14:51:28 GMT server-start server INFO 200 Wowza Streaming Engine 4 Subscription Edition 9.9.9.99dc build 20181127203650 - - - - 0.707
the 0.707 is seconds/milliseconds since it was started and each logging line should have the value in it.
Does that help you @Marco Bernasconi?