Log-normal cast values

I have a sampling process with some very small and some rather large random variables. As I have understood, there can be sampling issues with tiny variable sizes, (i.e. mu=1e-7, sd=1e-8)

To cope with this I have tried a log-casting. My assumption is that doing a tt.log/math.log operation on all input prior parameters, this size issue will be resolved. This however gives distributions with means and sd that are the log of the original values: mu=log(orig_mu) and sd=log(orig_sd).

After the sampling process i want to obtain my original non-log casted distribution. My initial thought was to do math.exp for all values, but that gives a log-normal distribution and not a normal distribution with parameters orig_mu and orig_sd,

Is there a way to do this in a good way? Is my logic reasonable?