How can I compute logarithm when comparison is undecidable?
Even though absolute comparisons may not converge, you should be able to narrow the argument into at least one of several partially overlapping ranges, such that you have a technique that works in that range.
For example, you should be able to tell that $x$ definitely falls into at least one of the ranges $A = \left(0,\frac{3}{4}\right]$, $B = \left[\frac{1}{2},\frac{3}{2}\right]$, or $C = \left[\frac{5}{4},\infty\right)$ with little difficulty. Use AGM if it's in $C$, the transformation if it's in $A$, and if it's in $B$, use this transformation:
$$\log (x) = \log (2x) - \log 2$$