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logarithms in the 21st century

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logarithms in the 21st century

Old 8th May 2022, 11:33
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logarithms in the 21st century

we have been doing some recent A level maths exam problems and revising logarithms. It seems that there is a concentration on problems involving logs in different bases. Not just the base 10 and base e that we had to deal with 45 years ago but any other integer bases as well. My wife has been helping a couple of students with these. We have worked out how to solve using our old calculators by converting to base 10, which our calculators do with ease. It was a bit of a surprise to find that the calculators used by A level students these days are able to work in any base you want, they have a button with log and two rectangular boxes marked against it. It was also a surprise to find that they don't operate in the same way that our old calculators do, requiring a different mindset and button presses to get the answer you want.

I do wonder when logs in any base other than 10 or e would be used. Anyone?

I'm not even going to try to understand the modern calculator's way of operating, HP RPN seems to be as good as any way for me. My current favourite is the "free42" programmable scientific calculator emulator which I have on my Android mobile phone, a Linus Mint laptop and a Win 8 laptop.

Rans6...............................

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Old 8th May 2022, 11:54
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I can't recall having met logs to any other base than 10 or e.
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Old 8th May 2022, 12:30
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Logs to base 2 are an inherent part of any binary arithmetic, and similarly base 8 for octal and base 16 for hex. Musical notes are a logarithmic scale to base 12 - something you need to know if you're making a new cherrywood fretboard for your beloved Stratocaster.

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Old 8th May 2022, 13:06
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No, music does not use base 12. The relevant base is 2^(1/12) if you can claim that decibels are based on 10^0.1.
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Old 8th May 2022, 13:37
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Originally Posted by Richard W View Post
No, music does not use base 12. The relevant base is 2^(1/12) if you can claim that decibels are based on 10^0.1.
The Mathematics of Music
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Old 8th May 2022, 13:53
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Originally Posted by rans6andrew View Post
It was a bit of a surprise to find that the calculators used by A level students these days are able to work in any base you want
...even a secret underground base?
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Old 8th May 2022, 14:12
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I expect this exercise is mainly to break the notion that these functions are just keys on a calculator, though the presence of keys to do exactly those functions sort of undoes that. On a slide rule one could see that multiplication by logarithm was adding distances and that division was by subtraction of distances. Then came calculators and multiplication and division are just buttons and external visualization was gone.

In using other bases it seems intended to say these functions are more than the common log and natural log, that any base could be used. I think one can use logarithms of imaginary and complex numbers. OTOH - it does seem like make-work to introduce a topic that mathematicians like (which is why they are mathematicians) and may never lead anywhere for the students. Anyone getting to powers with and of complex numbers will be long past simple base changes.

Base 2 is the most likely other use - this report may be a useful: What Every Computer Scientist Should Know About Floating-Point Arithmetic. It's such a key document that it is hosted in many locations.
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Old 8th May 2022, 15:04
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I remember using logs at school many years ago (the little book of log tables was standard equipment ), but I’m not sure that I ever used them when I left, even as an engineer.

I bought my first calculator a few years after I left school, which just did the basics. Dividing 0.0 by 0 used to put it into a loop it couldn’t get out of. The next generation ‘e’ was a huge improvement!
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Old 8th May 2022, 15:13
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As I was starting secondary school log books were being 'phased out' as calculators in the 80's were cheap as chips (did you know that in 1990 that 50% of the worlds computing power was in calculators? Now it is like 0.00000001%).

But I diverge... my post was really about Slide Rules!

Now never used one but they look magnificent... esp. the really expensive ones of the 70's.
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Old 8th May 2022, 15:24
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Originally Posted by MechEngr View Post
it does seem like make-work to introduce a topic that mathematicians like (which is why they are mathematicians) and may never lead anywhere for the students
Agreed and really a significant problem with our approach to education

In junior school, our headmaster said that school should prepare someone for adult life and I've often thought he was correct. To that end they should teach How to balance a bank account, How to use credit wisely, How to stay healthy, How to chose a car and how to drive it, How to plan a family, How to apply for a job, how to keep a job, How to stay out of jail, How to spot lies on the internet, How to save money, How to live within your means, How to stand up to peer pressure, How to think for yourself etc etc etc

But instead we learn a lot of esoteric information that admittedly has some value, like what are the countries of the world and where they are, but I still think we largely miss the point
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Old 8th May 2022, 15:41
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It has been my impression (steadily reinforced over the last 50 years) that A level maths is designed to test whether a pupil has the 'right sort of brain' to do a maths degree, but has very little relevance to "real" mathematics.

Maths Undergraduate Year 1, lecture 1. Forget everything you think you knew about maths, lets start at the very beginning. This is the empty set...........
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Old 8th May 2022, 16:16
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Originally Posted by rans6andrew View Post
we have been doing some recent A level maths exam problems and revising logarithms. It seems that there is a concentration on problems involving logs in different bases. Not just the base 10 and base e that we had to deal with 45 years ago but any other integer bases as well. My wife has been helping a couple of students with these. We have worked out how to solve using our old calculators by converting to base 10, which our calculators do with ease. It was a bit of a surprise to find that the calculators used by A level students these days are able to work in any base you want, they have a button with log and two rectangular boxes marked against it. It was also a surprise to find that they don't operate in the same way that our old calculators do, requiring a different mindset and button presses to get the answer you want.

I do wonder when logs in any base other than 10 or e would be used. Anyone?

I'm not even going to try to understand the modern calculator's way of operating, HP RPN seems to be as good as any way for me. My current favourite is the "free42" programmable scientific calculator emulator which I have on my Android mobile phone, a Linus Mint laptop and a Win 8 laptop.
Spoiler
 



Rans6...............................
This would just be the change of base formula for logarithms, E.g. log₃5=logₑ(5)/logₑ(3)=log₁₀(5)/log₁₀(3)

The calculator only needs to be able to do one log, and can use the change of base formula to use any other base.




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Old 8th May 2022, 17:20
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Originally Posted by Fitter2 View Post
It has been my impression (steadily reinforced over the last 50 years) that A level maths is designed to test whether a pupil has the 'right sort of brain' to do a maths degree, but has very little relevance to "real" mathematics.

Maths Undergraduate Year 1, lecture 1. Forget everything you think you knew about maths, lets start at the very beginning. This is the empty set...........
I did not know you were sitting in on my first maths lecture at University. Where were you sitting?
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Old 8th May 2022, 17:56
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The musical application of base 2 logarithms has to do with the way Western music divides an octave in to 12 "equal temperament" semitones:
  • Moving an octave up is a doubling of the frequency, regardless of where you start. 200Hz is an octave higher than 100Hz, 880Hz is an octave higher than 440HZ, and so on. Note vs Frequency is an exponential function.
  • So moving up a single semitone requires "1/12 of a doubling", you could say.
  • You want an operation that you can repeat 12 times to get a doubling of the frequency. If you have a multiplication function f, and you plan to repeat it n times, you can replace that with f^n.
  • In this case, n = 12, and f^12 = 2, so f = 2^(1/12).
  • Why 12? Because it's a pretty good match for natural harmonics. A "natural 5th" is a frequency ratio of 3:2 or 1.5 - that is, if you have an A = 440Hz, its natural 5th (E) will be 660Hz. It turns out that 7 semitones is very close to 1.5, since 2^(7/12) ≅ 1.49831. So a 5th (3:2) translates to a 7 semitone interval.
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Old 8th May 2022, 19:14
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Originally Posted by Saintsman View Post
I remember using logs at school many years ago (the little book of log tables was standard equipment ), but Iím not sure that I ever used them when I left, even as an engineer.

I bought my first calculator a few years after I left school, which just did the basics. Dividing 0.0 by 0 used to put it into a loop it couldnít get out of. The next generation Ďeí was a huge improvement!
I was using log tables and slide rules in anger within the aviation industry up until c1970 and only stopped at the time because it was quicker to learn Fortran, use a punch card machine and learn to load up the IBM1130 so that the whole two week set of manual calculations could be re-done overnight when a couple of parameters changed. So I trump your calculator with a computer!!

A year or so earlier I recall using a 64(?) step Hewlett Packard calculator and what I remember most was that it cost the same as an E-type Jaguar.


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Old 8th May 2022, 19:20
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like what are the countries of the world and where they are
They sure don't teach that in Murrica.
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Old 8th May 2022, 20:24
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I recall that a colleague in 1979/80 flew to New York on Freddie Laker's airline to buy a scientific calculator and reckoned this was cheaper than paying UK prices. I think consumer electronics always cost the same number of $ as £ even when the exchange rate was very different.
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Old 8th May 2022, 20:35
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Originally Posted by bnt View Post
The musical application of base 2 logarithms has to do with the way Western music divides an octave in to 12 "equal temperament" semitones:
  • Moving an octave up is a doubling of the frequency, regardless of where you start. 200Hz is an octave higher than 100Hz, 880Hz is an octave higher than 440HZ, and so on. Note vs Frequency is an exponential function.
  • So moving up a single semitone requires "1/12 of a doubling", you could say.
  • You want an operation that you can repeat 12 times to get a doubling of the frequency. If you have a multiplication function f, and you plan to repeat it n times, you can replace that with f^n.
  • In this case, n = 12, and f^12 = 2, so f = 2^(1/12).
  • Why 12? Because it's a pretty good match for natural harmonics. A "natural 5th" is a frequency ratio of 3:2 or 1.5 - that is, if you have an A = 440Hz, its natural 5th (E) will be 660Hz. It turns out that 7 semitones is very close to 1.5, since 2^(7/12) ≅ 1.49831. So a 5th (3:2) translates to a 7 semitone interval.
See post #5 for a simpler explanation.
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Old 8th May 2022, 22:29
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Is there a maths forum out there? What has this to do with a pilots site?
No offence intended, it's just well above me!
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Old 8th May 2022, 23:21
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Granddaughter is in year 9 (14 yo) and enjoys and does well at maths. Needless to say, she hasn't used log tables or slide rule, so I showed them to her as a point of interest. Likewise, when she tells me what she's doing in maths, I try to show her, from my experience, how it could be used in the real world.
When I was in high school, two of my maths teachers had been WW II bomber navigators. They had no difficulty finding real-world examples that interested a class of teenaged boys.
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