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We present the Cost-Distance problem: finding a Steiner tree which optimizes the sum of edge costs along one metric and the sum of source-sink distances along an unrelated second metric. We give the first known $O(\log k)$ randomized approximation scheme for Cost-Distance, where $k$ is the number of sources. We reduce several common network design problems to Cost-Distance, obtaining (in some cases) the first known logarithmic approximation for them. These problems include single-sink buy-at-bulk with variable pipe types between different sets of nodes, facility location with buy-at-bulk–type costs on edges (integrated logistics), constructing single-source multicast trees with good cost and delay properties, priority Steiner trees, and multilevel facility location. Our algorithm is also easier to implement and significantly faster than previously known algorithms for buy-at-bulk design problems.