We deal with the problem of the existence and uniqueness of a solution for onedimensional reflected backward stochastic differential equations with one lower barrier and with two strictly separated barriers when the generator has logarithmic growth y ln |y|| + |z| \sqrt{|ln |z||} in the variables y and z. The terminal value ξ and the obstacle processes (Lt)0≤t≤T and (Ut)0≤t≤T are Lp-integrable for a suitable p > 2. First, we construct the solution for the one barrier case using the localization method, and also the penalization method after we prove the comparison theorem in our setting. Then, we move on to the case of two barriers where we use the concept of local solution to construct a global one. As applications, we broaden the class of functions for which mixed stochastic control problems and mixed zero-sum stochastic differential games admit an optimal strategy