@anntzer Can you review this?

The remaining issue is that the formatting is lost after changing the scale from linear to log and back to linear again. Intuitively I feel like scale objects should be cached (most likely in a name->scale dict attached to the axis object) so that formatting is not lost... but perhaps this would be overengineering it? (Also, one would need a way to bypass the cache, most likely by passing an explicit scale object to set_scale.)

That definitely feels like overengineering to me. If someone did want to keep track of the full history of the formatters/locators then it should live in a layer above base mpl.

My gut feeling/expectation is that after creating a figure programatically, any changes done to it from the figure options UI should be reversible, which is why I proposed the scale object cache.

Where to put the no-op check comes down to what behavior you expect for something like

ax.set_xscale("log")
<mess with formatter options>
ax.set_xscale("log")

Ignoring the implementation right now, what do you think about this?

@tacaswell this appears to be waiting on feedback from you.

I would expect that given a set of artists on in the axes ax.set_xscale() should always put the figure back into the same state, independent of past history. Once we go down the route of remember this sort of thing we will end up turning every property into a time series with matching heuristics.

One thing that will help here is once we have traitlets, setting the default values of the properties on the LogFormatter and LogLocator will be easy.

That's fair. In this case, I'd suggest a comment in the definition of set_xscale and friends warning that setting the scale to its current value may undo other changes (possibly pointing to this diff as an example).