We consider a cognitive network consisting of n random pairs of cognitive transmitters and receivers communicating simultaneously in the presence of multiple primary users. Of interest is how the maximum throughput achieved by the cognitive users scales with n. Furthermore, how far these users must be from a primary user to guarantee a given primary outage. Two scenarios are considered for the network scaling law: (i) when each cognitive transmitter uses constant power to communicate with a cognitive receiver at a bounded distance away, and (ii) when each cognitive transmitter scales its power according to the distance to a considered primary user, allowing the cognitive transmitter-receiver distances to grow. Using single-hop transmission, suitable for cognitive devices of opportunistic nature, we show that, in both scenarios, with path loss larger than 2, the cognitive network throughput scales linearly with the number of cognitive users. We then explore the radius of a primary exclusive region void of cognitive transmitters. We obtain bounds on this radius for a given primary outage constraint. These bounds can help in the design of a primary network with exclusive regions, outside of which cognitive users may transmit freely. Our results show that opportunistic secondary spectrum access using single-hop transmission is promising.