In this paper, a cognitive multiple access channel in which the secondary transmitters seek to communicate with a common secondary receiver is considered. Before data transmission, it is assumed that cooperative spectrum sensing is performed with possible errors. For this channel, achievable rate regions are initially derived under two scenarios depending on how secondary transmitters access the channel. In the first scenario, secondary users can send data under both busy and idle sensing decisions by adapting the energy level of the transmitted signals according to the sensing result. In the second scenario, the secondary transmitters are not allowed to perform data transmission if the channel is sensed as busy. Subsequently, the performance in the low-power regime is analyzed by characterizing minimum energy per bit and slope regions in these scenarios. The impact of channel sensing performance (e.g., the probabilities of detection and false alarm, channel sensing duration) on achievable rate region, energy efficiency and slope region considering both scenarios are investigated.