Advanced Driver Assistance Systems (ADAS) form the basis of the autonomous driving. Adaptive Cruise Control (ACC), Lane tracking assist and collision avoidance systems etc. are some popular examples of active safety system technologies of ADAS. In order to prevent traffic accidents and ensure the security and comfort in the traffic, there are several intelligent vehicle technologies. Among them, the Adaptive Cruise Control (ACC) is one of which has crucial importance. In this technology, the adjustment of the speed of the vehicle is done by perceiving the environment and controlling through considering obstacles and other vehicle's positioning and speed. This technology involves the usage of high-tech sensor fusion and control techniques. The limitations of the environment are considered and if there is no danger, vehicle keeps its preset speed. If there is a vehicle in front, the vehicle reduces its speed by using dynamically determined equations that involves velocity and acceleration constraints. In this study, a fuzzy logic based adaptive speed control design has been proposed for low speeds. In practice, a fuzzy inference system was designed as a high level operation, and a simple mathematical model based on vehicle dynamics was used as a low level operation. The performance of the proposed system was tested on a scenario designed in a simulation environment and successful results were obtained.