just check what 'median' means. the median stellar mass is such that half of all stars have lower and the other half hihger mass. it has nothing to do with the logarithm. I havn't seen the term (median stellar mass) in the scientific literature.
Looking at the paper you're referring to, they never use the expression "median stellar mass". Moreover, this study is not concerned about the masses of individual stars, but with the total stellar mass of a galaxy, which they donote by $M$. They then consider the typical or characteristic value for $M$ (as obtained by fitting a Schechter function to the observed distribution of $M$), and denote it by $M^ast$. Since this is a rather large number, of the order of $sim10^{10}$M$_odot$, they prefer to use its logarithm to base ten: $$logleft(frac{M^ast}{mathrm{M}_odot}right).$$
I think the paper explains all that quite clearly and am puzzled by your difficulty.
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