I'm asking here to clear up a potential misconception or have an explanation on how exactly the oberth effect works.
context: rocket in deep space, no external forces on the rocket.
From what I understand, using the formula for linear kinetic energy        KE = m(v^2)/2 you can conclude that adding 100m/s while initially travelling at 1000m/s will add a great deal more energy to the ship than adding 100m/s while initially at rest after the initial energy is accounted for: (((1000+100)^2) - (1000^2))*M/2 >> (100^2)*M/2

And, from what I understand of Tsiolkovsky's rocket equation, this velocity I am supposedly adding is proportional to the exhaust velocity of the propellant relative to the craft, and the natural logarithm of the ratio of mass of propellant used.

In both cases where the craft is at rest or at an initial velocity, the ∆v is the same, and is dependant on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this ∆v.
So I'd wager the same quantity of chemical energy is converted to Kinetic energy, yet I'm left with this seemingly unexplained (in the example) 200,000 (2*100*1000) J/Kg more energy, and I'm clueless as to where it could have come from.

I've searched around the internet, resulting in explanations involving efficiency, but what about when the energy supposedly saved from the oberth effect exceeds the chemical energy stored in the rocket?