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.