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I am working on a data set and I have log-transformed two variables a and b and would like to figure out the relationship between the respective ratios. I have a strong intuition that a/b should have a well-defined range for a given value of lna/lnb, but my maths is poor.

a and be are positive integers with a finite range

0 < a < amax, amax=499639

1 < b < bmax, bmax=837481

I am looking for the range of

a/b

when

ln(a+1)/ln(b+1) = x

0 < x < 1

I know that for x = 1, a/b = 1 and when x = 0, a/b = 0

For anything in between there must be a range.

I'd like to plug in an x and get a range of possible values for a/b.

I have run a regression analysis on my data and figured out the deviations, but I am looking for a cleaner, more general solution that I could potentially also compute when the ranges of a and b are different.

Is this even possible?

Thanks in advance. I'd appreciate any help.