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I think this equation is derived using the Arrhenius equation for two different temperatures but it is not on the objectives sheet as something we need to do. Instead, I would memorize the equation because it is useful and not on the equations sheet.
If the original equation is the Arrhenius equation, what happens to the lnA when you derive the equation used to calculate rate constants at different temperatures (the one written in the original question)?
The Arrhenius equation can be re-written as ln k = -Ea/RT + ln A if you take its natural log. In this new form, you can compare the rate constants at two different temperatures using the equation you listed since subtracting lnk of one temperature and lnk of another cancels the term lnA.
This is done using two different symbolic temperature values, plugging them into the Arrhenius Equation, and subtracting these equations to solve for the rate constants. Pretty much the same as what we did for the Van't Hoff equation (relating temperature changes to a change in K)
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