New example image:

Maybe this is nitpicky, but could we have a "plain" semilogy plot next to the semilogx one? I think the base 2 labeling is super useful to include, but it takes some interpretation. It'd be nice to have a semilogy plot that you can see at a glance what's going on, and then the base 2 labeling plot could be kept but moved down.

It'd also be nice to have the minor gridlines turned on for one of these plots.

Indeed a bit nitpicky. 😄 It's a bit the question of how much can you illustrate with an example.

The original example had the base 2 on the loglog plot. I found this very unfortunate because (1) I've never seed double-log plots with different bases. That seems quite exotic. (2) to get that, one had to use set_xscale after loglog, which is not quite instructive as you override the x log scale defined by loglog. I thought it's nice to keep the base 2 aspect somewhere, and the semilogy was the one where it was least confusing. But maybe it's better to leave the base 2 out here?

I also think that adding minor grids here would be too cluttered.

If we want to have all the aspects in, we probably need to structure the example into subsections:

1. semilogx, semilogy, loglog
2. base 2
3. minor grids
4. log with errorbars that get negative

However you want to slice it! I think it's definitely an improvement over what's there currently.

@scottshambaugh I've reworked the example. There's now one plot with semilogx/semilogy/loglog, also containing minor grids, as that makes it more obvious which directions are logarithmic.

I've added separate examples base=2 and handling of non-positive values.

I think this looks great, breaking it up into sections was a good move and the exposition helps a lot to interpret what's going on. Good to merge unless you want another reviewer!