I am using a linear-log model to test whether overseas development assistance and remittances positively affect FDI in cases of good governance and financial market development. Let's say I want to interact the log of net official development assistance received (ODA) with governance. Governance is estimated as a score from -2.5 to 2.5. So I cannot apply the log transformation to governance, but can I still interact the log of ODA with governance? My hypothesis is that ODA will positively affect FDI in cases of good governance.
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Linear models (i.e., OLS regression) do not make assumptions about the distribution of explanatory / predictor variables, except that they are known constants. (To learn more about the assumptions of linear models, it may help you to read this thread: What is a complete list of the usual assumptions for linear regression?)
You are free to transform any, some, all, or none of your variables as you choose. Some reasons for choosing to do so are listed in this excellent answer by @whuber: In linear regression when is it appropriate to use the log of an independent variable instead of the actual values? The interpretation of log transformed variables is well explained in this thread: Interpretation of log transformed predictor.
Outside of these issues, the existence of an interaction, or the fact that the transformed predictor is included in an interaction, does not change anything.