could not open log file "/var/log/postgresql-2015-05-24_181212.log": Permission denied
I know that only root can write to /var/log ...


You can change that. As root user (or with sudo) in the shell:

mkdir /var/log/postgresql
chmod 0700 /var/log/postgresql
chown postgres:postgres /var/log/postgresql


And in postgresql.conf:

log_destination = 'stderr'      
logging_collector = on      
log_directory = '/var/log/postgresql'
log_filename = 'postgresql-%Y-%m-%d_%H%M%S.log'     
log_rotation_age = '1d'          # That's what makes it "dynamic"


Depesz has a detailed article on the topic:
https://www.depesz.com/2011/05/06/understanding-postgresql-conf-log/