Originally, closing conditions like that were use to induce stationarity by putting in a small risk premium. Schmitt-Grohe/Uribe (2003) showed this to not matter too much for linearized models with small parameter values. That may be different for nonlinear models, see https://www.bankofcanada.ca/wp-content/uploads/2017/06/swp2017-21.pdf
Finally, don’t do a full exp()-substitution. Append auxiliary variables. See