Yes, it's posisble to protect both the name and the logo of your brand with a trademark. You have to options of doing so:
- You can register two separate trademarks, one for the name and the other for the logo. This would give you the highest possible level of legal protection. However, it's more expensive and you have to ensure both the name and logo or eligible for registration.
- Alternatively, you may be able to gain protection for both your name and logo by registering a figurative mark containing both elements. This can be beneficial if your name is considered too descriptive to be registered as a wordmark. It is also a more cost effective option. However, you must then use the figurative mark as registered without any modifications. Also, you brand name itself will be protected to a lesser extent.
To understand the difference between registering wordmark, figurative mark or wordmark and figurative mark together, please have a quick read in the article Logo versus name: What defines your brand and thereby deserves trademark protection?