Where is the log file generated by a verbose start-up in OS X Lion?
Starting in at least Snow Leopard and possibly Leopard, all logging on the system goes through the Apple System Log (ASL) mechanism, which is like a central daemon and database for all log messages. The ASL daemon also makes sure that appropriate log messages that meet certain traditional criteria are written out to traditional flat text log files like system.log and kernel.log, but everything goes to the ASL database as well.
For more information, see the man pages for syslog(1), logger(1), syslog(3), asl(3), syslog.conf(5), asl.conf(5), syslogd(8), etc.
You can use syslog(1) to issue queries against the ASL database. Also, /Utilities/Console.app provides a GUI for generating and saving custom queries for the ASL database.
I'm pretty sure that the kernel's message buffer (traditionally viewed with dmesg(8)) gets dumped into the ASL database as soon as boot has proceeded far enough for syslogd to be running, so you shouldn't need to call dmesg directly, except through force of habit. Or if your system isn't booting far enough for syslogd to be running.
The accepted answer (by Spiff) was not helpful to me, nor is there a kernel.log any more.
What works for me (on macOS 10.12.6) is this command:
log show --predicate "processID == 0" --start $(date "+%Y-%m-%d") --debug
It shows all of today's kernel msgs, including the boot process. It needs some cleaning up, but at least it contains all I needed to find.
Open Terminal.app and run
sudo dmesg
(you'll be asked for your password)
This will display system messages from the last startup until now (you'll find the startup messages at the top of the output). You may try
sudo dmesg | more
this will output those messages page by page.
I myself wondering how to get the log most appropriate to the startup without access to the console/terminal. It crashed at startup but you can get access to the file system through the install disk (I had a dual linux boot available). The file is:
/var/log/kernel.log
If you can access the console within the operating system, it's better to follow the other suggestions and use the appropriate commands.