Yes, you can start a blog in just 10$ dollars and even you can start for free. But in this case, you will not have your own domain name. Domain name is e.g. yourname.com. If you want a blog with your chosen name then you have to spend just around 10$. It depends what domain name you choose. Yes, you just have to pay for your domain name only because I will tell you the way to have free safe and secure hosting.
Below are the details for your blog hosting source and the domain name source, from where you can start a blog in just 10$.
Platform
Choose Google's Blogger. No doubt, it is the best source for blogging and for a beginner it is wonderful. You don't need coding skills, in just a few minutes you can start your own blog.
Domain Name
Choose Godaddy. Because it is very easy to link your Blogger blog with Godaddy.
You are done. I am telling you, don't get confuse and search about more sources, just go with it and you will be happy. Just start a blog with Blogger.com in few minutes and start posting.
FYI, I made hyperlinks between the posts, by clicking on those you can get more details for the same.