I've actually solved this myself thanks to the help of @bruce-payette and @robdy for pointing me in new directions. Logging all commands, also from previous sessions, can be done from PowerShell itself.
Enter this command in PowerShell to recieve the path to a file where PowerShell saves the history accessible by using the up key:
(Get-PSReadlineOption).HistorySavePath
This will output something like this:
C:\Users\[username]\AppData\Roaming\Microsoft\Windows\PowerShell\PSReadline\ConsoleHost_history.txt
Open this file to see all saved commands from previous sessions!