To be alerted of activity on your account, make sure you have “Email me about account activity” enabled in your account settings. With this option enabled, you’ll get an email when someone logs into your account or when a new app is authorized.
To review recent activity on your account, scroll to “Active Sessions” a little further down on the account settings page. Here you’ll find the list of browsers and locations that have accessed your account over the past 30 days. If you see anything you don’t recognize, you can end that session immediately by clicking the gray X next to it.
If you do spot some suspicious activity, it’s always a good idea to change your Tumblr account password just in case someone else has gained access.
Note that the “Active Sessions” section shows activity when it’s coming from a browser (including mobile browsers), but not when you’re using the app. You can always view and manage your authorized apps on the apps settings page.
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