Species richness corrected for area -- should area be log-transformed?
Let me offer my answer even though I have not worked in ecology.

You are asking two questions if I understand you correctly:


Should you normalize to the area sampled?


The answer to that depends on whether the area sampled is a nuisance variable or not. Is it the case that you just happened to sample from areas with (largely) different sizes? Or do you rather expect or are interested in whether it may display some interaction with the environmental factors that you study? In the first case, normalize with clear conscience. On the latter, you should probably include the area sampled in your model. The reason for this is that by normalizing, you are assuming that the relationship between the response variable (number of species) and the predictor (area sampled) is fixed and has a slope of 1 irrespective of the variation of the rest of the variables included in your model. If that is not the case, you want to explicitly add the area sampled as a fixed factor in your model.


Should you log-transform the area sampled?


If the answer to the first question is that the area sampled is a nuisance variable then it does not matter whether you log-transform or not as now you are interested in a new measurement, species density. Hence, you may work with the untransformed variable which would simplify the interpretation of the results by a tiny bit. If the answer to the first question is that you want to investigate the interaction of the variable 'area sampled' with your other predictor variables, then you do not need to log-transform your variable as this is rarely needed, even if you use parametric inference. To summarize, you do not need to transform the variable 'area sampled'.