Well, it gives you a measure of relative distance around the median, but I'm not sure if it's the one you want. But taking e to the MAD of the transformed data does not give you the MAD of the original. Of course, if you want the MAD of the original data, you could just take that. Here is some R code. Anything after a # is a comment. E.g. with some mixed up distribution:
set.seed(1234)
x <- c(rnorm(100), rnorm(100,3,2), rbinom(100, 1, .5)) + 10 #Adding 10 to avoid
#negative numbers
mad(x) #1.43
mad(log(x)) #0.14
exp(0.14) #1.15
Does the original data "vary by 15% around the median"? I don't know. It depends on what you mean. But % is probably not the right way to look at it, because you can change that by adding a constant. Here is a plot of the density of the original data with lines at 10.96 and 15% below and above.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/8JJFM.png)