alt Hacker NewsThe term "baseless logarithm" is really nonsensical and using it would be a great mistake.
Nonetheless, where the author of TFA is correct is that logarithms are a single physical quantity, like length, area or volume, and that choosing the so called "base" is choosing the unit of measurement for logarithms.
Logarithms are included in the dimensional formulae of many derived physical quantities, e.g. for describing the attenuation or amplification of waves during their propagation, where one uses quantities like logarithm per length and logarithm per time.
Changing the "base" of logarithms modifies the numeric values of all derived physical quantities exactly in the same manner as changing any other fundamental unit of measurement, like the unit of length or the unit of time.
Like for any physical quantity, the complete value of a logarithm is independent of the unit of measurement, because it is the product between the numeric value and the unit of measurement. When the unit of measurement is changed, both the numeric value and the unit are changed and the product stays the same (i.e. the logarithm corresponds to the same ratio, regardless what base is used to compute a numeric value for the logarithm).
Nowadays, the unit of logarithms is normally chosen between the octave (binary logarithms), neper (hyperbolic logarithms) or bel (decimal logarithms).
The units of measurement for logarithms are not the bases, but the logarithms of the bases, which is why e.g. the value of the number "e", the base of the hyperbolic logarithms, is never needed in any computation. The only values that are needed are "ln 2" or its inverse "log2 e", which are used to convert the numeric values of logarithms when the unit of measurement is changed between those corresponding to binary logarithms and to hyperbolic logarithms (a.k.a. natural logarithms, but there is nothing more "natural" about hyperbolic logarithms than about any other kind of logarithms).
"Baseless logarithm" is not nonsencial. Given that:
a baseless logarithm is simply a family of functions with similar properties. Perhaps it might be clearer if the author said something like the "logarithm property" rather than "baseless logarithm" but that's nit-picking and debatable.As for changing the base changes the numbers, I have to wonder if you've done any advanced linear algebra or, more specifically, tensors. The whole point of a tensor is that it operates the same on an object regardless of the basis. Put another way, if a and b are two representations of the same object with different bases then T(a) and T(b) are equivalent if T(x) is a tensor.
My point is that any numbers are an arbitrary choice and they don't define the underlying structure. The author here is talking about logarithmic structure.
This btw is why you learn about different bases in linear algebra and converting between them. Or even polar coordinates vs cartesian coordinates (in high school, for some reason). They're priming you to learn about structure. You get to groups and learn that group A and B are isomorphic they have the same mathetmatical structure.
Even when the numbers change.