pipe-paneを使って幸せになりましょう。
tmux new \; pipe-pane "cat >> $PWD/log.txt"
ただし、これはセッションまるごと保存しようとするものであり、
必ずしもログとして嬉しくないかもしれません。
user@vultr:~/foo$ tmux new "foreman start; bash" \; pipe-pane "cat >> $PWD/log.txt"
# セッションを手動で止める
user@vultr:~/foo$ cat log.txt
23:16:54 web.1 | started with pid 11223
23:16:55 web.1 | [2017-08-29 23:16:55] INFO WEBrick 1.3.1
23:16:55 web.1 | [2017-08-29 23:16:55] INFO ruby 2.4.0 (2016-12-24) [x86_64-linux]
23:16:55 web.1 | == Sinatra (v2.0.0) has taken the stage on 5000 for development with backup from WEBrick
23:16:55 web.1 | [2017-08-29 23:16:55] INFO WEBrick::HTTPServer#start: pid=11223 port=5000
^C(← いらない)23:17:08 system | SIGINT received, starting shutdown
23:17:08 web.1 | == Sinatra has ended his set (crowd applauds)
23:17:08 web.1 | [2017-08-29 23:17:08] INFO going to shutdown ...
23:17:08 web.1 | [2017-08-29 23:17:08] INFO WEBrick::HTTPServer#start done.
23:17:08 system | sending SIGTERM to all processes
23:17:08 web.1 | exited with code 0
user@vultr:~/foo$ exit ← いらない