No, actually, you have a logo, which is, at best, an image that people associate with your brand. A brand is best defined as: the customer's perception of your company. It's not what you tell your customers about your company, it's what they tell you.Design a logo, write a tag line, test it with some focus groups, and viola! You have a brand.
You don't own your brand. Your customer owns your brand.
This is what makes the concept of "branding" so powerful. When your business strategy matches exactly the actual experiences of your customers, you have created an unstoppable marketing force.