I believe this is possible with the Script Runner plugin that @Renjith mentioned in your other question:
Isn't this a duplicate question which you raised at https://answers.atlassian.com/questions/50266/how-to-get-a-report-of-issues-where-a-custom-field-has-been-updated ?
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You can implement issue listener and update the date field in issueUpdated as specified in http://confluence.atlassian.com/display/JIRA/Listeners
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