I know that ln(a+b) can be rewritten as ln(a) + ln(1+b/a) as long as a>c.
Does this last requirement, which I don't really understand, prevent the use of such a tool in working out an equation that will be used across timeseries data (or say in a model), when it is generally unknown at any one timestep (that the equation is applied) whether a or c is greater?
For example, suppose I have this equation:
X = ln (R / (tan theta))
But then I realize that R really should be represented as R = l + p, and I use this equation to make estimates of x for each timestep where I receive a new l and p value. Problem is, I never know for sure, at each timestep, whether l or p is larger. And I need to take this equation and substitute it into other equations, in hope of simplifying two different equations - it seems not knowing in the future, when the resulting "simplified equations" are applied presents simplification from progressing.