/var/log/syslog on Ubuntu derivatives. You'll get a lot of noise in there, including tvheadend messages.
You can extract the latter with:
grep tvheadend /var/log/syslog
... that will show you lines such as these from my system:
Mar 18 09:27:16 revo tvheadend: START: initialising
Mar 18 09:27:17 revo tvheadend[25387]: charset: 71 entries loaded
Mar 18 09:27:17 revo tvheadend[25387]: dvb: Found adapter /dev/dvb/adapter0 (NXP TDA10071) via USB (480 Mbit/s)
Mar 18 09:27:17 revo tvheadend[25387]: dvb: Found adapter /dev/dvb/adapter1 (Sony CXD2820R) via USB (480 Mbit/s), Reports valid SNR values
Mar 18 09:27:17 revo tvheadend[25387]: dvb: Adapter /dev/dvb/adapter0 will run in filtered mode
Mar 18 09:27:19 revo tvheadend[25387]: dvb: Adapter /dev/dvb/adapter1 will run in filtered mode
...this is tvheadend detecting my two adapters.
To find the OS initialising them, it's a little less obvious. They're kernel messages, so you can strip them out with grep kernel syslog - but be prepared for a lot of unrelated messages.
Probably the easiest way if you're at all comfortable with the command line is to use vi, and search for the string "dvb". You're looking for code like this:
Mar 14 08:42:48 revo kernel: [ 7.294298] em28174 #0: 1 bytes requested from i2c device at 0xa0, but 0 bytes received
Mar 14 08:42:48 revo kernel: [ 7.294310] em28174 #0: board has no eeprom
Mar 14 08:42:48 revo kernel: [ 7.294320] em28174 #0: Identified as PCTV DVB-S2 Stick (460e) (card=80)
Mar 14 08:42:48 revo kernel: [ 7.294331] em28174 #0: v4l2 driver version 0.2.0
Mar 14 08:42:48 revo kernel: [ 7.335753] em28174 #0: V4L2 video device registered as video0
Mar 14 08:42:48 revo kernel: [ 7.335765] em28174 #0: dvb set to isoc mode.
Mar 14 08:42:48 revo kernel: [ 7.343619] em28xx: New device PCTV Systems PCTV 290e @ 480 Mbps (2013:024f, interface 0, class 0)
Mar 14 08:42:48 revo kernel: [ 7.343631] em28xx: DVB interface 0 found: isoc
... culminating in something like ...
Mar 14 08:42:49 revo kernel: [ 8.307825] usb 1-4: DVB: registering adapter 1 frontend 0 (Sony CXD2820R)...
Mar 14 08:42:49 revo kernel: [ 8.308896] em28174 #1: Successfully loaded em28xx-dvb
... which you can see correlates with the adapter message from tvheadend.
If these messages are okay then it suggests that the OS is initialising the card (or thinks it is, anyway), and tvheadend is responding accordingly. I suppose it might be a firmware/driver issue - what OS are you running (uname -a), what kernel?