With "non-paramteric test for correlation" you possibly are referring to Spearman's rank correlation?
In that case, there are no normality issues. In addition, a log transformation would change nothing.
The rank correlation considers not the original values, but their rank. E.g. "income" wíth n=4 values
1000, 980, 777, 200 would be used as ranks 4 3 2 1 in the calculations. This ranking will not change if
instead you use ln(1000), ln(980), ln(777), ln(200).