Does android keep a log of when it starts?
I think you have caught yourself out, in short, nothing you can do!
Have a look at this source that explains why, specifically in this section:
System Partition and Safe Mode

The system partition contains Android's kernel as well as the
operating system libraries, application runtime, application
framework, and applications. This partition is set to read-only. When
a user boots the device into Safe Mode, only core Android applications
are available. This ensures that the user can boot their phone into an
environment that is free of third-party software.

Keywords is free of third-party software
Edit:
When Android boots up, it keeps a cache of the logcat, in a temporary buffer that is reserved found in /dev/log. That buffer gets recycled when it hits the threshold, obviously, the bigger the threshold, the slower Android gets with the continual spamming into the logcat buffer hence kept to minimum - iirc, its around 64K:
#define DEFAULT_LOG_ROTATE_SIZE_KBYTES 16
#define DEFAULT_MAX_ROTATED_LOGS 4

Source: system/core/logcat/logcat.cpp
Do not ask how many lines that can be, as to each and to every app, its different. Not alone that, the logcat disappears upon reboot!