The evolution of wireless communication networks has introduced various applications that require massive device connectivity and high spectral efficiency. Non-orthogonal multiple access (NOMA) technique is one of the most promising technologies to perform efficiently data transmission. The NOMA technique can allocate the same resource block for two users by super-imposing signals. At the receiver, the signals are separated by performing successive interference cancellation (SIC) technique. For efficient data transmission, the fading and shadowing effects of channels also play a pivotal role. Many researches have considered Rayleigh, Rician, Nakagami-m, and other fading channels in various perspectives. In our paper, a system model based on a NOMA network with two users over log-normal fading distribution in the presence of channel estimation errors and SIC imperfections is proposed. The performance is analyzed in terms of outage probability and simulations are performed with the assistance of Monte-Carlo simulations. The obtained results shown the effectiveness in comparison with the traditionally used fading distributions. The same analysis is also performed in various scenarios of power allocation levels, target rates, and imperfections. The transmit SNR and power allocation of the users are important for efficient communication in any fading distribution as shown in this paper.