Can I offer wordpress blogs to my users?

I don’t know if this is possible. I have a site (portal plus forum), and this portal have the option to give my user blog if they want to. Because the simplicity of this portal’s blogs i want to offer a better kind of blog. I install wordpress and is everything that i heard it to my friends.
My blog i put in https://blog.mysite.com any chance to make this subdomain a ‘register site’ and all of ‘my’ user have the possibility of create the blog that they want? like

https://blog.mysite.com/user_nickname

If this is possible how an i do this?

Thanks, and sorry for my english.