A sensing policy for the restless multi-armed bandit problem with stationary but unknown reward distributions is proposed. The work is presented in the context of cognitive radios in which the bandit problem arises when deciding which parts of the spectrum to sense and exploit. It is shown that the proposed policy attains asymptotically logarithmic weak regret rate when the rewards are bounded independent and identically distributed or finite state Markovian. Simulation results verifying uniformly logarithmic weak regret are also presented. The proposed policy is a centrally coordinated index policy, in which the index of a frequency band is comprised of a sample mean term and a confidence term. The sample mean term promotes spectrum exploitation whereas the confidence term encourages exploration. The confidence term is designed such that the time interval between consecutive sensing instances of any suboptimal band grows exponentially. This exponential growth between suboptimal sensing time instances leads to logarithmically growing weak regret. Simulation results demonstrate that the proposed policy performs better than other similar methods in the literature.