$\log(1+X)$ is exponential with rate $\lambda$; two of them add to Gamma$(2,\lambda)$

Let $X_1,X_2$ have the pdf $f(x\mid\lambda)=\dfrac\lambda{(1+x)^{\lambda+1}}$, $x>0$ (independently). The distribution of $Z=\log(1+X_1)+\log(1+X_2)$ is: