Recently James Bach posted a nice blog entry in which he identified logging as a good friend of an exploratory test team. I believe the comprehensive logging of AUT (application under test) is an extremely valuable asset to any tester regardless of the context of exploratory testing.
A detailed log will surely help to minimize the effort required to reproduction of issues. There are intermittent failures, random application crashes which are extermely hard to regenerate in successive attempts. If the AUT is written in a way in which all the major user events are logged, then the root causes of such random failures can be traced easily.
A comprehensive log is a useful test report. After a test run, you can store the log somewhere safely and use it as the test report. Specially, in agile test processes, test teams do not find sufficient time to record test results and maintain detailed test reports due to the short release cycles. If the AUT provides good set of logs, it will help testers to use them instead of maintaining separate test logs.
IMHO, comprehensive logging is a must have feature of any of the enterprise-scale application. At WSO2, this has been identified as a extremely important tool and the necessary modifications are in-progress to improve logging in WSO2 Carbon SOA middleware suite.