A law firm is a company that seeks to represent its clients in court and also offers them legal advice where required. Since this is a lucrative business and mostly targeted towards adults, the color, text, and design style of a law firm logo has to follow accordingly.
The design style of your law firm logo should be simple, professional, and elegant. Since a firm like this deals with a lot of documents and usually has business cards, the logo should translate well when used on any medium. There's no need for any quirky fonts or symbols, though there might be a little creativity when it comes to the actual styling. For instance, you may use negative spaces to create the impression of scales within your text.
The colors of a law firm logo would send certain messages to their audience. For example, a dark color would symbolize endurance, sophistication, and professionalism. The same goes for the font used, which should be formal but not boring. Comics Sans isn't recommended here, while the Herald font would present a decent appearance. It's easily readable and also distinguished, making it a perfect fit for a law firm logo.
Just a few generic symbols that you may include in your logo include a hammer and gavel, a pair of scales, or even a judge's wig. However, you may also consider including symbols to represent your particular specialization. For example, if you usually focus on cases regarding domestic violence, you can have a house included with your generic law symbol. This would help your specific audience to instantly pay attention even before reading the text or tagline.
If you're looking for the right logo design for representing your law firm, consider getting a professional logo designer on board. This would help you get the perfect logo for all your advertising needs.