Large organizations emit terabytes of logs every day in their cloud environment. Efficient data science on these logs via text search is crucial for gleaning operational insights and debugging production outages. Current log management systems either perform full-text indexing on a cluster of dedicated servers to provide efficient search at the expense of high storage cost, or store unindexed compressed logs on object storage at the expense of high search cost. We propose LogCloud, a new object-storage based log management system that supports both cheap compressed log storage and efficient search. LogCloud constructs inverted indices on compressed logs using a novel FM-index implementation that supports efficient querying from object storage directly, removing the need for dedicated indexing servers. Experiments on five public and five production log datasets show that LogCloud can achieve both cheap storage and search, scaling to TB-scale datasets.