An upper bound is derived on the capacity of a cognitive radio system by considering the effects of path loss and log-normal shadowing simultaneously for a single-cell network. Assuming that the cognitive radio is informed only of the shadow fading between the secondary (cognitive) transmitter and primary receiver, the capacity is achieved via the water-filling power allocation strategy under an average primary signal to secondary interference plus noise ratio loss constraint. Contrary to the perfect channel state information requirement at the secondary system (SS), the transmit power control of the SS is accomplished in the absence of any path loss estimates. For this purpose, a method for estimating the instantaneous value of the shadow fading is also presented. A detailed analysis of the proposed power adaptation strategy is conducted through various numerical simulations.