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As we know the rocket equation,

$$ \Delta v = v_e \ln\left(\frac{m_i}{m_f}\right) = I_{sp} \ g \ \ln\left(\frac{m_i}{m_f}\right) $$

So do $I_{sp}$ and the mass ratio have an inverse relation, or is it that $I_{sp}$ is inversely related to natural log of the mass ratio?

Does it mean that to have logarithmic relation implies in the short interval (early phase) the difference is very large?

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  • $\begingroup$ What do you mean by "in the short interval (early phase)"? There is no reference to time in the equation. Do you mean in the first part of a launch when the payload will be a very small fraction compared to the propellant+rocket structure? Also - what do you mean by difference? Difference between who? (I'm trying to clarify your question, although Russell Borogrove have probably already provided a good answer) $\endgroup$
    – BlueCoder
    Commented Nov 13, 2018 at 11:07

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$I_{sp}$ is inversely related to log of mass ratio if delta v is held constant, yes, but that's not how the rocket equation is usually applied.

The way the rocket equation is usually applied is that you have a delta-v requirement given by a particular mission -- for example, the 4100 m/s needed to get from low Earth orbit to lunar orbit. Your $v_e$ will be constrained by your available choices of rocket engine, and $m_f$ will be constrained by the payload you're trying to move.

Rearranging the equation to solve for $m_i$ is left as an algebraic exercise.

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