There are plugins that do what you are wanting. Just search for custom logo and many will come up for you to choose from.
I am trying to do something similar, but also allowing login via facebook, twitter, etc. I was using Social Login for WordPress to allow facebook login. But then also used a plugin to change the login page by changing the logo and adding a photo, but social login then stopped working.
If you are not wanting login with facebook, it should be easy. If anyone has done both successfully, I would be interested in which two you used. I have very limited coding skill, so plugins are better for me.