In Analyze Variability, you assess the log residuals because Minitab always uses the log transformation to model the standard deviation response.
Log residuals are the difference between the natural log of the observed response standard deviation and the natural log of the fitted standard deviation. They represent the part of the response that is not explained by the model. Of the types of residuals Minitab calculates in Analyze Variability, the log residuals most closely resemble regular residuals.
Use log residuals in residual plots to assess the fit of your model.
Standardized log residuals are the log residual divided by its (asymptotic) standard error. Minitab calculates the standardized log residuals using the method you select to analyze the variability of your model. If you use the least squares estimation method, the standardized log residuals are the standardized residuals obtained from doing weighted least squares regression on the log of the standard deviation of your response.
Use standardized log residuals in residual plots to assess the fit of your model.
Ratio residuals are the ratio of the observed response standard deviation to the fitted standard deviation. They represent the portion of the response that is not explained by the model. You can obtain the ratio residuals by exponentiating the log residuals. Residual plots using ratio residuals can be difficult to interpret.