There are many fairly successful businesses out there without a logo. You really don’t need a logo, unless you want to increase your profitability.
Within your community there are a lot of great opportunities for low-cost marketing, like sponsoring a high school sport, band, etc. Not only are you getting a little good PR, you’re helping a good cause. Sponsoring children’s and teen activities can lead to great respect in the community and top-of-the-mind awareness in your hopeful clientele. By having a logo, you have a reusable mark that can be used on t-shirts, coffee mugs, yearbooks, posters, and more. That mark reminds your community that you support them and they will in turn support you.