The Sybase ASA engine has a series of database startup flags that enable logging and enable you to limit the size of the active log file.
The following command uses the -zs flag to restrict the size of the active log to 10K.
Once the log reaches 10K, the mydatabase.log file is renamed mydatabase.old, and a new mydatabase.log is created.
(Any existing mydatabase.old file is overwritten.) This process is repeated in perpetuity.
dbeng9 -zr all -zs 10 -zo mydatabase.log
Consult the Adaptive Server Anywhere Database Administration Guide for additional information on Sybase ASA logging parameters.