This paper deals with risk-sensitive control problems and their associated Bellman equations. Using probabilistic and analytic methods, the author obtained the following results, under mild conditions. Here, he used the logarithmic transformation of the exponential criterion as a pay-off function. 1) The associated Bellman equation has a non-negative classical solution, less than the value function. This yields no break down of the value function. 2) Long time asymptotics of the solution is characterized by the Bellman equation of ergodic type control. 3) The large deviation principle is presented as an example.