This is something that we simply have to agree to disagree on. It depends on the field (i.e. context), which base is the most common. The weight of history also comes into play. As do the choices made by the authors of most common textbooks and computer algebra systems.

My personal preference is to always use 'log' with a subscript, 'lg' for base 10, 'ln' for base e, and 'lb' for base 2 (though I might also write log

_{2} for that). But this is just a preference.

If I'm coerced to make a pick, I would prefer 'log' to mean base 10. Admittedly that is because during my formative years calculators where not at all common, so folks learned to work with logarithm tables well before they learned anything about derivatives, and at that point base 10 was the obvious choice. In addition to the pH-scale, base 10 is used in telecommunications, because both error probabilities and signal-to-noise ratios are often described in powers of ten. The former because it is then easy to quickly come up with ball park figures, of how often something goes wrong - the latter because it is measured using the decibel (dB) scale.

Of course, in the context of complex analysis 'log' surely means base e, or rather, (one of the branches of) the inverse of the complex exponential function.

I frankly don't see the point of trying to compress the names of these functions in spoken language. There is usually enough background noise, and my goal is then to minimize the chance of anyone mishearing it. May be my bias to lecture room setting shows here

? I would simply use phrases like "natural log(arithm)", "Briggs' log" or "log base 10" and "binary log" or "log base 2". In particular when to the chalkboard. The audience immediately also gets an idea of my preferences making communication in the immediate future that much less prone to errors.

That was the tl;dr; version. The summary:

it all depends on the context.