There are various opinions about when you should and should not use these two. The one I was taught was that spearman was used for ordinal data and pearson for interval (pearson can not address non-linarity, spearman will work with non-linear data if it is monotonic in nature that is you have one rather than multiple directions for the curve).
I have not heard that logs work to eliminate outliers - they stabilize variance. There are robust methods to address outliers, but they are not simple (such as M estimators).