I don't know of any way to have ASE automatically log that to the errorlog.
If this is a general ongoing requirement, I'd recommend using the auditing features.
To show it historically, the transaction log in the master database can provide proof (assuming you weren't using a shared login). When you drop a database, ASE logs a BEGINXACT log record that has a transaction name of $dbdestroy <databasename>. The record also contains the suid of the user and a datetimestamp.
dbcc log(dbid,0,0,0,number of records to print,0)
The final 0 is the filter for BEGINXACT record types, which has an op code of 0.
Example
1> create database x
2> go
CREATE DATABASE: allocating 1536 logical pages (24.0 megabytes) on disk 'master'
(1536 logical pages requested).
Database 'x' is now online.
1> drop database x
2> go
1> set switch on 3604
2> go
Switch 3604 ('print_output_to_client') is turned on.
All supplied switches are successfully turned on.
1> dbcc log(1,0,0,0,1000,0)
2> go
LOG SCAN DEFINITION:
Database id : 1
Log operation type BEGINXACT (0)
Forward scan: starting at beginning of log for 1000 log records
Maximum of 1000 log records.
LOG RECORDS: (0 records did not qualify so far.)
BEGINXACT (4054,1) sessionid=4054,1
attcnt=1 rno=1 op=0 padlen=6 len=72
odc_stat=0x0000 (0x0000)
loh_status: 0x0 (0x00000000)
masterxsid=(invalid sessionid)
xstat=XBEG_ENDXACT,
spid=17 suid=1 uid=1 masterdbid=0 dtmcord=0
name=$dmpxact time=Mar 5 2015 9:26:25:270AM
BEGINXACT (4054,11) sessionid=4054,11
attcnt=1 rno=11 op=0 padlen=5 len=64
odc_stat=0x0000 (0x0000)
loh_status: 0x0 (0x00000000)
masterxsid=(invalid sessionid)
xstat=XBEG_ENDXACT,
spid=17 suid=1 uid=1 masterdbid=0 dtmcord=0
name=x time=Mar 5 2015 9:26:53:280AM
BEGINXACT (4054,19) sessionid=4054,19
attcnt=1 rno=19 op=0 padlen=2 len=72
odc_stat=0x0000 (0x0000)
loh_status: 0x0 (0x00000000)
masterxsid=(invalid sessionid)
xstat=XBEG_ENDXACT,
spid=17 suid=1 uid=1 masterdbid=0 dtmcord=0
name=$dbdestroy x time=Mar 5 2015 9:26:57:280AM
Total number of log records 3 qualified out of 3 scanned.
DBCC execution completed. If DBCC printed error messages, contact a user with
System Administrator (SA) role.
So the third BEGINXACT record output here shows that suid 1 ("sa", that is me), destroyed (dropped) database named "x" at 9:26AM
Cheers,
-bret
