Modelling curves using logarithms...

apple2357

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Mar 9, 2018
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So, i am a bit confused by this topic. I can see it works but don't understand exactly WHY?

So, suppose we have (x,y) data and we think it can be modelled by y= ax^b. And we want to try and work out values of a & b

One approach is to take logs of the data and re-plot it and then the data appears to lie on a straight line, we work out the gradient and intercept of this new data and can relate it back to find values of a& b.

This works because log y = log a+ blogx . So log a is the intercept and b is the gradient.

I get all this and can accept it. But

- What does taking logs of data actually mean? And why does it produce a straight line? Does it somehow rescale the axes, so the data lies on a straight line? I can see by the algebra above, it will work. But i can't quite work out what is going on under the surface? Can we just arbitrarily take logs of both sides, could we just take sine or both sides, or cosine?!!! etc

Or am i just asking the wrong questions?
 
So, i am a bit confused by this topic. I can see it works but don't understand exactly WHY?

So, suppose we have (x,y) data and we think it can be modelled by y= ax^b. And we want to try and work out values of a & b

One approach is to take logs of the data and re-plot it and then the data appears to lie on a straight line, we work out the gradient and intercept of this new data and can relate it back to find values of a& b.

This works because log y = log a+ blogx . So log a is the intercept and b is the gradient.

I get all this and can accept it. But

- What does taking logs of data actually mean? And why does it produce a straight line? Does it somehow rescale the axes, so the data lies on a straight line? I can see by the algebra above, it will work. But i can't quite work out what is going on under the surface? Can we just arbitrarily take logs of both sides, could we just take sine or both sides, or cosine?!!! etc

Or am i just asking the wrong questions?

"Why" can be a hard question to answer!

Your idea of rescaling the axis is one good way to look at it; if you plot the data on log-log paper (that is, with horizontal and vertical distances being the logs of the coordinates), the curve should be a straight line. This transformation makes it easier to see the relationship. I suppose you could, in principle, do the same thing with other functions, but you would want to do it because it turned equations of some form into linear equations to make it useful.
 
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