Today's graphs used in domains such as machine learning or social network analysis may contain hundreds of billions of edges. Yet, they are not necessarily stored efficiently, and standard graph representations such as adjacency lists waste a significant number of bits while graph compression schemes such as WebGraph often require time-consuming decompression. To address this, we propose Log(Graph): a graph representation that combines high compression ratios with very low-overhead decompression to enable cheaper and faster graph processing. The key idea is to encode a graph so that the parts of the representation approach or match the respective storage lower bounds. We call our approach "graph logarithmization" because these bounds are usually logarithmic. Our high-performance Log(Graph) implementation based on modern bitwise operations and state-of-the-art succinct data structures achieves high compression ratios as well as performance. For example, compared to the tuned Graph Algorithm Processing Benchmark Suite (GAPBS), it reduces graph sizes by 20-35% while matching GAPBS' performance or even delivering speedups due to reducing amounts of transferred data. It approaches the compression ratio of the established WebGraph compression library while enabling speedups of up to more than 2x. Log(Graph) can improve the design of various graph processing engines or libraries on single NUMA nodes as well as distributed-memory systems.