The logarithmic Sobolev inequality has been extensively studied in analysis and probability. After the equality case is fully characterized by E. Carlen, it is natural to ask how far an admissible function that is close to achieving the equality deviates from the optimizers. This question is called a stability for the logarithmic Sobolev inequality. In this talk, we discuss stability results in terms of the Wasserstein distance and the total variation distance. This is based on a joint work with Emanuel Indrei.