Use the transformation $z=\ln x$ to solve

$$\ddot{z}=-\dot{z}^{2}-1-e^{-z}$$

subject to the conditions $z=0$ and $\dot{z}=V$ at $t=0$, where $V$ is a positive constant.

Show that when $\dot{z}(t)=0$

$$z=\ln \left(\sqrt{V^{2}+4}-1\right)$$