I have original problem: \begin{equation} \begin{split} \max_{x\geq 0} &\; x \\ \;\text{s.t.}&\; x\leq a \\ \;\quad & \;\; \log(1+x)\leq b \\ \;\quad & \;\;a,b\geq 0 \end{split} \end{equation}
Solution: $x^*=\min\left(a,e^b-1\right)$.
I transform this problem using $\log(1+x)=t$ to have \begin{equation} \begin{split} \max_{t\geq 0} &\; e^t-1 \\ \;\text{s.t.}&\; e^t-1\leq a \\ \;\quad & \;\; t\leq b \\ \;\quad & \;\;a,b\geq 0 \ \end{split} \end{equation} Is this problem itself a convex? How can I show that?