The idea takes quite a bit more than logarithms. This one needs Taylor Series.
The Linear approximation, then, , simply discarding all the higher order terms. If x is close to zero, this is not an unreasonable approximation. I would check that stipulation before jumping in with both feet! (Note: Since these are all interest rates, or differences of interest rates, it seems a rational process.)
I'll simplify your notation a bit, using IFC and IUS for the interest rates.
Also, I realize there is a point to be made by writing the ratios that way, but is wonderfully simpler to handle.
That's all the background.
Now, it's time for the logarithms...
That was the easy part! This can be slightly rewritten, using log rules:
We are almost ready to use the linear approximation. The only problem is that pesky F/S term. It doesn't quite look like what we want. Let's just beat it up until it does.
Is this starting to look familiar? We now have:
This lends itself directly to the linear approximation of all three terms, simply
This is now quite easily solved for IFC - IUS. I'll leave the algebra to the interested and motivated student.