However, what is the relationship among x1, x2, and x3? I'm trying to think of a mathematical situation where you would be subtracting successive logs beyond the case of log x - log y.

log(2/3) = log 2 - log 3

If you have x1/x2, and then divided by x3, or x1 divided by x2/x3, then your equation would make sense. You could do log(2,x1/x2) - log(2,x3), or log(2,x1) - log(2,x2/x3).