Yes. Sure. The key is to understand that in the expression "linear regression" the word "linear" means "linear with respect to the coefficients in front of variables". So, you can not only transform each variable differently, but, for example, make two different transformations of each variable and include both in regression. You should keep in mind, however, that ideally your variables should be uncorrelated with each other and roughly on the same scale.
If you are using R, you can transform variables directly in the formula without changing the data frame
lm(y ~ I(log(v1)) + I(v2^2) + I(1/v3), data=data)
In this case, if you want to make predictions for another data frame
newdata, you can use it directly (without changing it).
Alternatively you can transform variables "by hand" in the data frame (introduce new columns with or without eliminating the old columns) and work with new variables
lm(y ~ new_v1 + new_v2 + new_v3, data=data)
To make predictions in this case, you need to transform variables in your
newdata data frame (that is, introduce new columns) in the same way you transformed it in
The results of these two "implementations" will be the same. And both are linear regression.