Answer
$$y = - \ln \left( {2 - x} \right)$$
Work Step by Step
$$\eqalign{
& {\text{Let the differential equation be }}\frac{{dy}}{{dx}} = {e^y} \cr
& {\text{Separate the variables}} \cr
& {e^{ - y}}dy = dx \cr
& {\text{Integrate both sides with respect to }}x \cr
& - {e^{ - y}} = x + C\,\,\left( {\bf{1}} \right) \cr
& \cr
& {\text{Find the function }}y = f\left( x \right){\text{ with }}y = - \ln 2{\text{ when }}x = 0 \cr
& - {e^{ - \left( { - \ln 2} \right)}} = 0 + C\, \cr
& - 2 = \,C \cr
& \cr
& {\text{Substitute }}C = - 2{\text{ into equation }}\left( {\bf{1}} \right) \cr
& - {e^{ - y}} = x - 2 \cr
& {\text{Solving for }}y \cr
& {e^{ - y}} = 2 - x \cr
& \ln {e^{ - y}} = \ln \left( {2 - x} \right) \cr
& y = - \ln \left( {2 - x} \right) \cr} $$
