If your LaunchAgent plist, you could add (for example) this to log to say /tmp/
<key>StandardErrorPath</key>
<string>/tmp/net.syncthing.syncthing.stderr</string>
<key>StandardOutPath</key>
<string>/tmp/net.syncthing.syncthing.stdout</string>
Now, I use Homebrew too (awesome) but for something like SyncThing, it really doesn’t make much sense to use it in my opinion. Homebrew is great for things with lots of dependencies/options, say php, but SyncThing being a single binary it seems overkill. Just my opinion.