Where can I find the Tomcat log written out by 'grails run-app'?

There isn't a default log file, the output to the log gets written to stdout.

You have to define an log4j root logger in your Config.groovy like this:

log4j = {
    appenders {
        // console name:'stdout', layout:pattern(conversionPattern: '%c{2} %m%n')
        // file name:'file', file:'app.log'
    }

    // By default, messages are logged at the warn level to the console and the app.log
    root {
        warn 'stdout'
        // warn 'stdout','file'
        additivity = true
    }
...
}

This example also shows how to configure the logging pattern. Also it shows how to configure file logging. The appenders section is optional and just needed to configure the logging pattern or file logger.