Logarithmic Scale: Calculating Evenly Spaced X Values

[Jdo300]

Hello All,

I am working on a program that produces plots that use a logarithmic scale on the X-Axis (for showing decade frequency ranges). The Y-axis can have any arbitrary value and I'm not thinking about that at this point, but here is my question.

Say, for simplicity that every Y-value on the graph was set to a value of 10 so that a straight line is always drawn. Is there a way to determine the X coordinates such that they appear to be evenly spaced going from left to right on the graph even though the X-axis is displaying a logarithmic scale? I want to know this so that I can generate a plot that has x number of evenly spaced points. The second challenge is I want to be able to specify a starting and ending point on the X-axis to draw the graph (it may not always start at 0). How would I design a formula that if I enter the starting X value, end X-value, and number of points, that i could spit out point n in the range to plot a function with?

Thanks,
Jason O

 

[AUMathTutor]

I think so.

Say you're doing the logarithms to the base of 10. Then to get evenly spaced x-coordinates when the x-axis is on a log scale, just start with a number (1, for instance) and multiply by 10 over and over again. So

x = 1, 10, 100, ..., 10^n

Taking the Log gives you:

Log x = 0, 1, 2, ..., n

Which, I believe, is exactly what you wanted.


Say you wanted them spaced at intervals of 2 rather than 1. Then multiply by 100 each time. Say you wanted 1/2. Then try sqrt(10). Etc.

 

[Jdo300]

So Far here is what I have come up with so far, only this function works for a linear scale and not logarithmic:

x(n) = (n(End - Start) / Count) + Start

where:

n = position in the array of points
End = Ending X coordinate
Start = Starting X coordinate
Count = Number of points in the array

I basically want to figure out how to convert this function to the other scale.

- Jason O

 

[Jdo300]

Hi AUMathTutor,

That sounds like the right idea. But I'm not sure how to implement it when you have arbitrary starting and ending points along with n number of points specified.

Thanks,
Jason O

 

[AUMathTutor]

Alright, so you want "count" many evenly-spaced points between "start" and "end".

First, get the log base 10 of start. Call it log_start. Then get the log base 10 of end. Call it log_end.

Next, generate evenly spaced points using the same method you posted above, but call the array log_x(n). So we have...

log_x(n) = (n(log_end - log_start) / Count) + log_start

Then, just go through the array and set

x(n) = pow(10, log_x(n))

 

[AUMathTutor]

For instance, let start=10, end=10,000, and count=4. Then...

log_start = 1
log_end = 4

log_x(0) = log_start
log_x(1) = (log_end - log_start)(1/4) + log_start = (1/4)log_end + (3/4)log_start
log_x(2) = (log_end - log_start)(2/4) + log_start = (1/2)log_end + (1/2)log_start
log_x(3) = (log_end - log_start)(3/4) + log_start = (3/4)log_end + (1/4)log_start
log_x(4) = (log_end - log_start) + log_start = log_end

so...

x(0) = start = 10
x(1) = (end)^(1/4) + (start)^(3/4) = 15.623413251903490803949510397765
x(2) = (end)^(1/2) + (start)^(1/2) = 103.16227766016837933199889354443
x(3) = (end)^(3/4) + (start)^(1/4) = 1,001.7782794100389228012254211952
x(4) = end = 10,000

There you have it. You'll have to iron out issues related to whether you want count or count+1 points, but still. The idea is there.

 

[Jdo300]

Thank you very much! I'll try this out and let you know what I end up with .

 

[Jdo300]

Hi AUMathTutor,

I finally had a chance to test out the formula you gave me. It works like a charm! Thanks!

- Jason O