Can you use logistic regression if you have more than two options for the dependent variable. Instead of just a yes/no option I could have more than 2 options. I have 20+ predictor variables.
If you have more then two categories it's not logistic regression, but multinomial logistic regression (also known as multinomial regression). While logistic regression model is defined in terms of conditional odds of one category as compared to another, in multinomial model you compare $K-1$ categories to the $K$th category that serves as a "default" category. If your categories are ordered, you will use ordinal regression, that is a similar model but for ordered categories.
See the Interpreting exp(B) in multinomial logistic regression thread for extended discussion of multinomial model.