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One problem, for example--was to calculate the change in entropy of a given sample as it gained heat per temperature.

[tex]\Delta S=\int^{f}_{i}\frac{dQ_{r}}{T}[/tex]

The dQr is assumed to be relatively reversible for the problem, and you end up solving the problem by recognizing Q as mc(delta)t.

[tex]\Delta S=mc\int^{f}_{i}\frac{dT}{T}[/tex]

Integrating yields mc(ln(Tf)-ln(Ti)), and then you recognize the properties of the natural logarithm and divide Tf by Ti, take the natural log, and finish the problem.

The point where I get screwed up logically is that the units don't come out correctly if you don't treat the natural logarithm this way, and simply compute through subtraction. I know its a property of the natural logarithm to divide, and I've gotten stuck on problems like this in the past (simple rocket fuel propulsion examples, etc). My main question is--does this happen every time a natural logarithm is involved after integration? Are there cases where the units

*don't*cancel like this? I know its a silly question, but it has been bugging me all night.