I have a doubt regarding standardization, I have to use a multivariate regression and one of the variables is in log-scale. Is using standardization enough to re-scale the log variable, or is it necessary to apply a transformation to a linear scale and then apply standardization?
There are many ways to do feature scaling, and it's hard to know which one is better (if you find any white paper on this please comment below). I would say that if you have outliers, nonlinear transformers such as applying the log are good options, but there are many others (see this great sklearn documentation page on this).
Regarding your question, it depends. You must do some exploratory analysis to check the variable's distribution on each configuration. If you see that your variable's histogram is "good enough" (in terms of a well-defined distribution -- sorry to use too general language), then go ahead and use it without further standardizations.