The short answer is, you can do this any way you want. But how you do it will set the tone for the story.
If you write it in a log format, it will bring to mind military experiences or a formal log for a job (like a medical record).
For example, here is a World War II era U.S. Naval log. The starting location is given for each day and each entry has the time in 24 hour format minus the standard colon. The date is given with the day then the day-month-year which is military and international.
(I'm including headers from the page that you wouldn't necessarily put in a book...though if you did, it would be just the first time, if it was from the same log.)
U.S.S. CRUX (AK-115)
c/o Fleet Post Office
San Francisco, Cal.
Wednesday, 19 April 1944
Moored to berth #1A, Norfolk Navy Yard, Portsmouth, Virginia.
1015 — Loaded two thousand (2000) rounds 20mm ammunition.
1430 — Loaded six hundred (600) rounds 40mm ammunition.
A medical record uses the time if there are multiple entries per day (such as in a hospitalization) and it's usually in military format. The date can be abbreviated or written out. The location is in the header.
June 3, 2019, 1320 hours
A diary or letter format will bring to mind a single character setting her/his thoughts to paper. In the US, the standard is month-day-year. The day of the week and the time are optional and up to the writer. A location is generally only there if it's different from the usual places the author writes, like if s/he is on vacation.
June 3, 2019. Monday, 1:20pm.
Monday, Jun 3, 2019. Camping at the American River
For a standard "here's where the characters are right now!" format, just give the facts in a non-military format. Use 12 hour time if you're American (Europeans use 24 hour time even if not in the military). I would write out the date so it's easy to read, and use the American format (again, unless you're not in America). I would not put "in" and other elements to create sentence structure. This is factual information, not a narrative. And don't spell out numbers. Don't say "USA" if you're giving a state, unless the readers aren't in the US and don't know the book is set there (or if only one scene is set there). There's also no need to put the state if it's a famous city like LA.
Sunday, June 2, 2019. 9pm. Los Angeles.
Jun 2, 2019, Sunday night, Los Angeles.
Los Angeles, 9 PM. Sunday, June 2, 2019.