KELVIN scale-Can't make sense of the last step of this solution

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The discussion centers on the transition from the equation ln T_c - ln T_H = θ_c - θ_H to θ = ln T + C, questioning why the subscripts dropped out. It explains that by applying the exponential function, e^(θ_c - θ_H) equals T_c/T_H, leading to the logarithmic form ln(T_c/T_H) = ln T_c - ln T_H. The simplification occurs because ln e equals one, allowing the equation to reduce to ln T_c - ln T_H = θ_c - θ_H. The terms "C" and "H" represent different temperature measurements, but when T_H is considered fixed, only one variable remains, making the distinction unnecessary. The conversation highlights confusion over the assumption of T_H being fixed in the context of the problem.
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I do not understand how they went from \ln T_c-\lnT_H=\theta_c-\theta_H
to
\theta=\ln T+C

Why did the subscripts drop out? What just happened?
 
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exp means e^thethac if that is thetha

so you get e^(thethac-thethah) is Tc/Th.

If you ln both sides, you'll get ln(Tc/Th) = lnTc-lnTh and on the other side lne^(thethac-thethah) = (thethac-thethah)lne. lne is one.

Thus

lnTc-lnTh = thethac-thethah
 
"C" and "H" just represent different measurements of the temperature. Since you take TH to be a fixed temperature, that just leaves one temperature- you don't need the "C" to distinguish it form "H".
 
I am sorry Halls, I don't quite follow. What do you mean I take TH to be fixed?
 

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