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In the context of statistical physics, critical phenomena are accompanied by power laws having a singularity at the critical point where a sudden change in the state of the system occurs. In this work, we show that lean blowout (LBO) in a turbulent thermoacoustic system can be viewed as a critical phenomenon. As a crucial discovery of the system dynamics approaching LBO, we unravel the existence of the discrete scale invariance (DSI). In this context, we identify the presence of log-periodic oscillations in the temporal evolution of the amplitude of dominant mode of low-frequency oscillations $(A_f)$ exist in pressure fluctuations preceding LBO. The presence of DSI indicates the recursive development of blowout. Additionally, we find that $A_f$ shows a faster than exponential growth and becomes singular when blowout occurs. We then present a model that depicts the evolution of $A_f$ based on log-periodic corrections to the power law associated with its growth. Using the model, we find that blowout can be predicted even several seconds earlier. The predicted time of LBO in good agreement with the actual time of occurrence of LBO obtained from the experiment.