2.1.8.7 ✓ Mathematica. Time used: 0.003 (sec). Leaf size: 17

ode=D[x[t],t]==b*Exp[x[t]]; 
ic={x[0]==1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 1-\log (1-e b t) \end{align*}