Yes, generally when dealing with logarithms, they deal with unitless operations (exception is when looking at pH where . The connotations in doing these operations are implied).

Looking at first-order reactions, you know that the equilibrium constant k has a unit of . So, rearranging as you did:

Inside the operation of the logarithm, the units cancel out and on the right hand side, the units cancel out as well (given that time is in seconds).