What kind of technique was used in this linearized log rate?

Unless there's additional context here, the k' is just a name for "log k + m log$B$".   Think of it as useful shorthand.

You don't have to use a new name/variable.   You could instead have that last line be:

log rate = (log k + m log$B$) + m log$A$

But if you're going to have to keep typing (log k + m log$B$) over and over, it's easier to just make up a new variable like k' and write k' + m log$A$.

Most of the time in math if you see a line that starts with the word "let",  like "let k' = log k + m log$B$",  it means that the thing on the left is being introduced for the first time and will be used as shorthand for the stuff on the right.
Is this an extinction rate, equilibrium constant, reaction constant or pH calculation arrived at during a chemical reaction equation? Just curious. I just dealt with those last semester.

I agree that the "prime" in "k' " is the notation used in ordinary differential calculus for the first derivative of a function k. It was confusing to me, too, when I came across it in a general chemistry class.

I think these calculations do involve derivatives and exponentiation at some point in the math happening "behind the scenes," because they are rates and deal with chemical reaction speeds and logarithmic values of ionic hydrogen concentrations in solutions. However, I think they are just alluding to it here or "telling you,"  without proving it mathematically. I asked similar questions and got "that's a different class, where you go over the derivation of these formulas."

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