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        riddles >> putnam exam (pure math) >> Logarithmic Graph Paper
        
(Message started by: Foolish on Dec 1^st, 2004, 11:01am)

Title: Logarithmic Graph Paper
Post by Foolish on Dec 1^st, 2004, 11:01am

I was thinking earlier today about graphing things on logarithmic paper and semi-logarithmic paper.

Say you graphed a circle, radius r on a sheet of standard 1:1 graph paper, then transcribed it onto semi-log paper, what would the area and circumference become? How about on true logarithmic paper? And then, a sphere in log space (volume and surface area)?

This is perhaps too easy a problem for this forum, but I thought that it was too mathy for the easy or medium sections.

Another thought, what would the shape of the transcribed circles become if they were drawn back onto the 1:1 paper?

Title: Re: Logarithmic Graph Paper
Post by Aryabhatta on Dec 1^st, 2004, 10:47pm

On a log paper, if we want to plot the function f(x) do we actually plot ln(f(x)) where ln is the natural logarithm ?

Assuming that we want to log-plot the quadrant sqrt(1-x²) for the area we are looking for the definite integral:

₀[int]¹ ln[sqrt](1-x²) dx

The substitution x = sin[theta] and using integration by parts seems to do the trick, but we might have problems as ln[sqrt](1-x²) is unbounded in the interval [0,1) (I haven't actually tried to find the value).

I am not sure if that was your intent.. or I understood you correctly.

Title: Re: Logarithmic Graph Paper
Post by Foolish on Dec 1^st, 2004, 10:59pm

I will try to clarify what I mean:

-regular graph paper's grid has consistent hash lines of 1 unit each, for both the x and y axis.
-semi-log graph paper has regular hash markings for the x axis, but the y axis has 1, 10, 100, 1000...
-full log paper has the 1, 10, 100, 1000... for both x and y axis

basically, we are just changing the values of the lines on the graph paper, without changing how we draw the shape.

I will make some images if that would be more helpful...