When issuing the sh log command, it usually gives me logs from june to present. It takes quite some time to go through them to reach the latest logs.
Is there a method or command which allows me to display the logs for just today or for the past week...
I believe that you may get pretty close to what you are looking for with the "show log | include" feature. The pipe include gives you the ability to match a character string. So if you were to enter
show log | include Aug 24
then it would show only records in the log that have Aug 24 in the log record. This presumes that you have configued your syslog to use timestamp datetime.