Fractals represent the phenomena of expanding symmetries which exhibit similar patterns for different scales. In this paper, we establish an escape criteria by using Dogan and Karakaya (DK) iterative process to generate fractals namely Julia and Mandelbrot sets for the logarithmic function F (z) = log(1 + zp) + c , where c ∈ ℂ and p ≥ 2. Our result is a generalization of the existing algorithm and technique providing fractals for different parameter values. Also, the time taken to obtain fractals for different parameters by using computer software MATLAB is computed in seconds.