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In this correspondence, we consider a code-division multiple-access (CDMA) channel with a linear receiver structure whose inputs are subject to a total power constraint. Each user is required to satisfy a certain quality-of-service (QoS) requirement expressed, for instance, in terms of data rate or delay. The set of all feasible QoS requirements is called the feasibility region. It is shown that if the signal-to-interference ratio (SIR) at the output of each linear receiver is a bijective and log-convex function of the QoS parameter of interest, the minimum total power needed to satisfy the QoS requirements is a jointly log-convex function of the QoS parameters. Furthermore, in two special cases of practical interest, we show that the minimum total power is strictly log-convex. These results imply that the corresponding feasibility regions are convex sets. The convexity property is a key ingredient in the development of access control strategies for wireless communications systems.