hi

on Tails as on any other Linux Distro, temporary files are mainly written in /tmp, /var/run and /var/tmp. and somewhere in your account folder (here: /home/amnesia), depending on the application and its settings.

log files are written in /var/log.

All temp and log files don't survive after shutdown: they are written in the RAM (in fact, with Live Systems - without persistence feature - all the system or the applications need to write is written in the RAM).

Maybe you don't understand what a Live System is: the system is not modified by a session. It is exactly the same before and after. Perssitence is an additional feature that allows the user to keep some files across reboots, but they are stored in a separate area; in the case of Tails, on an encrypted partition. What is really persistent on a Live System is the Live System itself.

If you need or are curious to know what the Live System does, what is the content of the temp and log files during a session, open a terminal, type 'df' (obviously without the quotes), and open a filemanager to browse the directories (the last column of the 'df' output) for which the first column of the 'df' output does not begin with a slash (/).