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In a Generalized Linear Model (GLM) with a log link function, the coefficients are interpreted in terms of their exponentiated values. Each coefficient represents the change in the log of the expected value of the response variable for a one-unit increase in the predictor variable, assuming all other variables in the model are held constant.

When you exponentiate the coefficient value, you transform the log-odds or log-linear relationship back to the original scale of the dependent variable. This allows for interpretation in terms of multiplicative effects. Specifically, if the exponentiated coefficient is greater than 1, it indicates that an increase in the predictor variable results in a percentage increase in the expected value of the response variable. Conversely, if the exponentiated coefficient is less than 1, it suggests a percentage decrease in the expected value.

This interpretation as percentage changes is crucial for understanding the effect sizes in the context of the original data, making the results more intuitive and actionable. The choice to exponentiate the coefficients thus provides a clear, meaningful understanding of how changes in predictors affect the response variable within the framework of the GLM.