What do you mean by unstructured? Do you mean the data has yet to be parsed into a format which could be logically grouped into columns? Or do you mean that it's deeply nested?
Since log data doesn't really change, it might be overkill to use something like HBase (or any database for that matter). On the tools team at Cloudera, we've found that writing the data into HDFS and using Impala to analyze it works pretty well.
We typically analyze chunks of log data and then ingest it into HDFS (due to the use case), but if you're looking to ingest data in "real-time", you'll want to use something like Apache Flume.
With the data separated into partitions, we're able to run queries that analyze GBs of data in under a second (15 nodes). This is log data (LOG4J) that has been extrapolated into columns, and then loaded into a columnar storage format (RCFile, soon to be Parquet).
Let me know if you have any questions, glad to help.