Testing Difference of Values from Lognormal Distribution?

#1
I have data that fits a lognormal distribution really well. My primary goal is to create a statistical classifier using this distribution to apply to new data from the same source. Essentially, I have 2 possible values, I choose the bigger one (based on the lognormal density) and then attach a statistical test score to that maximum probability.

As part of the output of this classifier, I would like to know the statistical significance of the difference between two values from the lognormal. For normal data, the z-score and Standard Error can be used, but is there an equivalent method for the lognormal?
 
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