So my understanding thus far is that if there is no trend, then any arithmetic or logarithmic differencing will still yield a stationary time series.
If the series is arithmetically increasing, then either a differencing or logarithmic adjustment will yield a stationary time series.
If the series is geometrically trending, then only a logarithm can make the series stationary.
My reasoning is that you could add zero or multiply by a logarithm with the base equal to the stationary mean, and still have get the stationary mean as the correct answer.
I'm thinking of it as (and correct me if I'm wrong):
Geometric Trending Adjustment = Square
Arithmetic Trending Adjustment= Rectangle
No Trend Adjustment = Quadrilateral
Such that any type of adjustment can be made to a no-trend adjustment, but a geometric trending adjustment requires a more specific type of adjustment.