How to write log message in QGIS's script algorithms GUI tab?

You can do this with global variable progress. see the user manual, processing chapter

progress.setText('Youpi')

Will print 'Youpi' in the processing log.

It supports html formatting:

progress.setText('<b>Youpi</b>')

to print in bold...

progress come also with setPercentage() methods to control the progress bar

For QGIS 3.x you can use the pushInfo() method on a feedback object:

feedback.pushInfo('This is a log message')

See the scripts section of the user's manual for other relevant methods.

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