[MaeveChondrally (OP)]

Least Percent Error
F = sum from i=1 to N,  (1 - (m*ln(xi)+b)/ln(yi) )^2
diF/dim = sum ( 1 - (m*ln(xi + b)/ln(yi) )*(ln(xi)/ln(yi) ) = 0
diF/db = sum (1 - (m*ln(xi) + b)/ln(yi) )*(1/ln(yi) ) = 0
b = 1/sum(1/ln(yi) )^2*[-sum(1/ln(yi) + m*sum(ln(xi)/(ln(yi) )^2 ]

m=[ sum(1/ln(yi )^2*sum(ln(xi)/ln(yi) )  - sum(1/ln(yi) * sum(ln(xi)/(ln(yi))^2]/[ sum(1/ln(yi) )^2*sum( ln(xi)/ln(yi) )^2   -  ( sum ln(xi)/(ln(yi) )^2 )^2

where m is the slope of the linear logarithmic  line of least percent error among the data points
and b is the y intercept of the linear logarithmic line of least percent error.

this equation is useful in chemistry and astrophysics and image processing and MRI contrast enhancement for finding
a line that is resistant to quantum noise and works over many decades or scales where a line of least square error would be useless.  Especially, this might be generally and theoretically more desirable for all data sets than least squares error.   the distribution of error about each xi can be found with application of software found here:
https://library.wolfram.com/infocenter/MathSource/9086/ [nofollow]
This was first discovered by myself in 1982 , but may have been seen elsewhere before or since. i did use it throughout university too from 1983 to 1988.

[Bored chemist]

I once asked a statistician about this. Why use least squares rather than least linear distances? and the answer is that least squares is the calculation that gives the best fit to the data based on the assumption that the data distribution is normal.
Least linear distances is the best fit if the error distribution is triangular (like the sum of two dice).
I think the "least ratio" approach you are looking at is equivalent to doing a least squares analysis on log transformed data.
In that case it's related to this
https://en.wikipedia.org/wiki/Log-normal_distribution
That's the distribution commonly used as a model for concentration of pollutants in air. It no doubt has other uses.