Welcome to the JavaRanch, Rabindranath!
Tomcat's console is not actually a log file, just the stdout and stderr output from the Tomcat JVM. The stock scripts for running Tomcat usually redirect it to catalina.out, but that's just a convention and not actually an architectural constraint. Which is one reason why we tell people that webapps should do system.out.println - you never know where the print output may be sent.
So
you should be able to create a "log file" by starting Tomcat with a "catalina start >C:\TOMCAT_HOME\logs\catalina.out" (adjust path to fit your preferences). Since the pop-up window is provided by Windows, if you redirect both stdout and stderr ("2>"), Windows shouldn't pop up that second window, since it's only provided when a stdio output needs a place to present itself.
There are other ways of running commands in Windows that avoid the second window as well, but I'm out of practice on them.
The secret of how to be miserable is to constantly expect things are going to happen the way that they are "supposed" to happen.
You can have faith, which carries the understanding that you may be disappointed. Then there's being a willfully-blind idiot, which virtually guarantees it.