HP Solve Newsletter Jan 2011: Article about LOGs



#2

The article about logarithms was very inspiring (HP Solve #22, 42-50). 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

HP-calculators 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 French-American 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


#3

Very interesting post. Let me add this:

Quote:
The result is that the first fraction of log10(x) is always higher than the integer part of the respective number.

This is true for x > 1,371. Below that point 10 lg(x) is less than x.
Quote:
...estimation of log10(600) would result in a fraction somewhat higher than "6", maybe "2.7" or "2.8".

I'd like to add another method for estimating common logarithms - a very simple method if you're familiar with the world of photography. Due to the logarithmic property of light (our eyes see the same brightness difference if the light intensity changes by the same factor (!)) the usual aperture scale is a geometric sequence where each value is sqrt(2) times its predecessor. In other words: the common logs of the aperture values change in 0,15-steps. Which means that logs can be determined easily by simple couting:
aperture value   1    1,4    2    2,8    4    5,6    8    11
common log 0,00 0,15 0,30 0,45 0,60 0,75 0,90 1,05
In this case lg 600 is easily estimated to be a bit more than 2,75.



It gets even more precise if you're familiar with intermediate values, for instance 1/3-steps, leading to nice 0,05 logarithmic steps. So, knowing that the next 1/3 aperture step after f/5,6 is f/6,3 the unknown lg 600 has to be quite exactly in the center between 2,75 and 2,80 - and in fact it is (2,778).



Okay, this is a method specially designed for photo nerds. But I'm sure there're some of us around here... :-)
Quote:
In order to manage log10 one would have to learn by heart the logarithms within one order of magnitude

That's what most photographers have learned from the start - maybe without realizing it. They use geometric sequences like apertures, shutter speeds and ISO-settings and at the same time they think in logarithmic exposure stops (EV).



Dieter


Edited: 21 Jan 2011, 8:57 a.m.


#4

Quote:
I'd like to add another method for estimating common logarithms - a very simple method if you're familiar with the world of photography...

Okay, this is a method specially designed for photo nerds. But I'm sure there're some of us around here...

That's what most photographers have learned from the start - maybe without realizing it. They use geometric sequences like apertures, shutter speeds and ISO-settings and at the same time they think in logarithmic exposure stops (EV).


Dieter,

Indeed, I am one of those photo nerds. As I read Frido's post, I was thinking about this same subject, so thank you for spelling it out so well.

#5

Thanks for the tip on relating f-stops to logs.

I suppose that photo nerds (love the term!) have as many cameras and/or lenses as they do calculators!

The digital age has caused an explosion in my number of cameras, at least. For almost 40 years, I got by with my Pentax ME film camera and 3 lenses (28, 50, and 135 mm). (Reminds me of the durability of my HP 35 from the same era!)

In the last 5 years or so, my digital cameras included HP210, HP318, Canon XT, Canon XTi, Kodak 915, Kodak 981, Pentax K100D, Pentax K10D, and Pentax K20D. Since old Pentax lenses work on the new Pentax DSLRs, I have also acquired some 15-20 lenses (on the shopping site which must not be named).

I imagine there are a lot of other photo nerds here. Tell us what you have.

Related question: what (perhaps high-priced) item do you own whose operating lifetime is measured at a few hundred seconds?


#6

Quote:
Related question: what (perhaps high-priced) item do you own whose operating lifetime is measured at a few hundred seconds?

Chateau d'Yquem 1959?

#7

Quote:
Related question: what (perhaps high-priced) item do you own whose operating lifetime is measured at a few hundred seconds?

The CCD on your digital camera?

#8

I am thinking of Martin's answer (include the entire camera, though), although I also like Chris's answer - but he must drink his Chateau D'Yqeum fairly rapidly!! (I am looking forward to savoring my Chateau Lafitte-Rothschild 1996, that my wife bought for me back when it was affordable, for my 65th birthday in a few weeks! I think I will spend more than a few hundred seconds at that.)

Consider your camera: average exposure is perhaps 1/100 second; multiply by 20000 or 30000 pictures, and the operating life is several hundred seconds (although shutters on high-end DSLRs are typically rated for 100K or more exposures - so you gain a few more hundred seconds). If you are into time exposures, you gain a lot of "lifetime."

#9

CCD sensors on almost all DSLRs now potentially have a much longer operating life, given that they now are used for "live preview" mode as well as for video capture.

Even on DSLRs that don't have those features, the short operating life you describe is only due to the way the device is conventionally used, and not due to any limitation of the sensor. The MTBF of the sensor is in the hundreds of thousands of hours. Some photographers use cameras for very long timed exposures.


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