In the textbook, Introductory Econometrics (Wooldridge), the concept of specifying log/log, level/log ... , is explained as a tool to understand the relationships of the regerssors either in relative or absolute terms. Using log transformation is also explained as having desirable properties like capturing non-linear relationships in a linear regression, reducing the influence of outliers and so forth.
Question: My question mostly refers to the point on log capturing non-linear relationships. If log transforming captures non-linear relationships, why are there examples in the textbook and in the real world where we have both log and squared or cubed terms? something like:
log(y) = c + log(x1) + log(x1^2) + log(xk) + u
Wooldridge also points out ambiguity in what log squared terms mean. It could be as above, or the one below (where we log first, then square the whole term):
log(y) = c + log(x1) + log(x1)^2 + log(xk) + u
In either event, is it fair to say that log transforming alone does not adequately capture non-linear components, and thus we include squared/cubed terms? Or, perhaps using log is mainly for theory (relative/absolute inference) and including squared/cubed terms still is needed?