In this paper a portfolio optimization problem with transaction costs Is studied. Transactions of assets are formulated as an impulsive control, which does not allow continuous transactions. The introduction of fixed rate costs has the effect of preventing continuous transactions. The objective of this paper is studying the problem of maximizing the growth rate of expected log utility. A quasi-variational inequality (QVI) of "ergodic type" is derived from the optimization problem. To solve the inequality, we use a perturbation method, where we obtain a necessary estimate of solutions of non-ergodic type QVIs by using a stochastic representation of the solutions.