Comment by adrian_b

You use the word "logarithm" with the meaning "logarithmic function", i.e. a function whose argument is a ratio and whose result is a numeric value that gives the corresponding logarithm in a certain base.

I use the word "logarithm" in its original sense, meaning "logarithmic quantity". Logarithms are a certain kind of quantity, which measures numeric ratios, like other quantities measure various things, e.g. plane angles, lengths, time or cardinal numbers, where the latter measure how many elements are in a set.

Even for cardinal numbers, where there is an obvious "natural" unit, the number "1", it is frequent in practice (e.g. when computing statistical quantities) to choose other units of measurement, like a thousand, a million, a billion, the Avogadro number, the Curie number, etc.

Both for a logarithm or for a cardinal number, like for a distance or an angle, the complete value is independent of the chosen unit of measurement, even if the numeric value changes.

As you say, while for a scalar quantity the complete value is independent of the unit of measurement, for a vector quantity or tensor quantity the complete value is also independent of the chosen reference system of coordinates, even if the numeric values of the components of a vector or tensor change when the reference system is changed.

However, all these have nothing to do with whether the term "baseless logarithm" makes sense.

You say that this should be used as a term with the meaning "logarithmic function" (because the family of functions defined by you is the same as the family of functions traditionally named "logarithmic functions", since Leonhard Euler).

I say that this claim is baseless itself, because the term "logarithmic function" has been in use for almost three centuries and there is absolutely no need to invent another term, which also does not make sense etymologically, because when computing any logarithmic function, i.e. any member of the function family that has the property mentioned by you, you need a concrete base value, i.e. no such function is baseless.