In this paper we consider the power utility maximization problem under partial information in a continuous semimartingale setting. Investors construct their strategies using the available information, which possibly may not even include the observation of the asset prices. Resorting to stochastic filtering, the problem is transformed into an equivalent one, which is formulated in terms of observable processes. The value process, related to the equivalent optimization problem, is then characterized as the unique bounded solution of a semimartingale backward stochastic differential equation (BSDE). This yields a unified characterization for the value process related to the power and exponential utility maximization problems, the latter arising as a particular case. The convergence of the corresponding optimal strategies is obtained by means of BSDEs. Finally, we study some particular cases where the value process admits an explicit expression.