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For rockets, thruster, I found the Tsiolkovsky rocket equation:

$$\Delta v=v_e ln(\frac{m_0}{m_f})$$

Of course, rockets can travel up to relativistic speeds, thus a relativistic implementation of the RE must be taken into account for a higher degree of accuracy in calculations.

All implementations I found, rely on this equation which gives the mass ratio in terms of ##\Delta v## and exhaust speed ##v_e##... but I couldn't manage to derive it, since in all papers it is given without demonstration:

Could you please give me a hint on how to get to this identity?