Thanks to the Publicize module, you can automatically publish each one of your blog posts to your Tumblr blog.
To get started, go to the Settings -> Sharing page, then click the Connect button next to the Tumblr logo.
You’ll be taken to Tumblr where you’ll be asked to approve the connection between your site and your Tumblr account. Click Allow.
If you have more than one Tumblr blog you will then be prompted to choose which blog to publicize to.
You can also make the connection available to all users of the site. Meaning that when another user creates a post it will also publicize to this Tumblr connection.