Proving A(xy) = A(x) + A(y) without Calculus

[wofsy]

let A(x) be the area under the graph of 1/x from 1 to the number,x,where x is bigger than 1.

Can you show without using calculus or any of the properties of logarithms that A(xy) = A(x) + A(y)?

 

[breedencm]

So how are we supposed to define "area". It's clear how to define area for triangles, rectangles, etc... But how do you suggest we define area for this particular curve? Usually, area is defined as the integral.

 

[wofsy]

So how are we supposed to define "area". It's clear how to define area for triangles, rectangles, etc... But how do you suggest we define area for this particular curve? Usually, area is defined as the integral.

yes you are correct. Maybe there is no way to do it. But still I wonder. Suppose you were trying to explain this miraculous property of 1/x to someone who did not know calculus.

I have always found the proof of the formulas for the log to be unintuitive.

 

[Hurkyl]

If we assume "area" is meaningful, and behaves correctly under rescaling, then the crux of the integral manipulations amount to little more than rescaling vertically by some factor and horizontally by its inverse, I think.

 

[g_edgar]

Probably you need to show: the area (under the graph 1/x) between 1 and a is the same as the area between b and ab ... so you need to show that the first area is transformed to the second if you stretch horizonally by b and vertically by 1/b. So you would need to know that such a transformation preserves area.