Hanson (2003, 2007) proposed the use of the logarithmic market scoring rule (LMSR) for eliciting private information about a future, verifiable, event. A market maker sets a baseline probabilistic forecast of the event and subsequent market participants report their own forecasts. Each participant is paid the logarithmic score of each of their forecasts, and pays the logarithmic score of the previous forecast. I show that the LMSR admits a Perfect Bayesian Nash equilibrium in truthful strategies in a setting in which agents receive information dynamically over several periods. This generalizes previous results in which information arrival is static, strengthening the status of the LMSR as an attractive payment scheme for prediction markets and forecasting tournaments.