HP Solve Newsletter Jan 2011: Article about LOGs Message #1 Posted by Frido Bohn on 21 Jan 2011, 4:52 a.m.
The article about logarithms was very inspiring (HP Solve #22, 4250). It summarized some basic truths in a very concise
manner and gave a historic overview starting with logarithmic tables, over slide rules up to calculators  especially
HPcalculators of course!
The concept of logarithms appears not easy to be approached in our linear cognition of quantities. Thus, it is not trivial
to estimate the log10 of, let's say 600 without any aids. We know, that it must be something with a "2" as the integer part,
as the number can be expressed as 6*10^2. But what about the fraction? Will it be closer to "1" or to "9"?
If we look at a linear representation of a graph of log(x) we see that the curve starts with a steep slope which decreases
as x progresses. The consequence of this is that the fractional part of log10 is higher at the beginning of the first decade
such that log10(2)= 0.30xx, whereas log10(5) = 0.69xx and log10(9) = 0.95xx.
The result is that the first fraction of log10(x) is always higher than the integer part of the respective number. So, the
estimation of log10(600) would result in a fraction somewhat higher than "6", maybe "2.7" or "2.8". As a matter of the
nature of log10, the fractional part will be identical in each order of magnitude such that log10(6 000 000)=6.7782 and
log10(600)=2.7782. In order to manage log10 one would have to learn by heart the logarithms within one order of
magnitude, and that's all. But how about log2 or ln (on the basis of e)? I believe that the approach to the logarithm of
other bases than 10 is very hard in our decimal world.
Related to the concept and perception of logarithms, I found it very interesting to read an article about the Mundurucu,
an indigene culture in the Western Amazonas. A FrenchAmerican group of researchers investigated the mapping of numbers
onto space in this culture and compared it to the American. The Mundurucu have no nouns for numbers beyond five.
Thereafter, they use relative quantities as "few", "more" or "many". The mapping of quantities on a line
(without labels) followed a logarithmic pattern whereas the Americans made it the "linear way". The researchers
concluded that our intuitive approach to quantities is rather logarithmic than linear but that we learn that the space
between, for example "4" and "5" must be the same as for "8" and "9", and thus we get implanted to think in linear
patterns as school kids. Indeed, experiments with children between 4 and 6 years of age show that the logarithmic
mapping of quantities is also common in this period of human development.
From an evolutionary point of view, the logarithmic approach of quantities appears to be more intuitive. For our
ancestors, it was important to distinguish between 2, 3 or 4 lions but it was less important to keep apart 84 from 85 or 86
gnus. In higher quantities, the distinction between 100, 200 or 300 have been of more relevance. Our perception of
sound and shades, for example, follows also a logarithmic mapping and is described in Webers's law  "larger numbers [or a
more intense sensation of sound or light] require a proportional larger difference in order to remain equally
discriminable".
In conclusion, logarithms always existed. They were not just invented by John Napier. He discovered them and made them
accessible to logical reasoning. Just as Newton did not invent gravity, but he was the one who put its concept into
physical laws.
Regards
Frido
