Log Tables

lostmyway said:

Maybe your mind does what it does whether you are looking or not.

I think minds do that.  They can do all sorts of things while you're not paying attention, it seems.

NAS22687 said:

[quote][/quote]

Yeah, but at least you can write the code in the first place!

Took me long enough, though!  Nothing I've written is mathematically clever, though I do have my moments of... well, stuff that is compact and efficient, apparently.  I admit I'm guilty of the occasional instance of "drunk coding" in the early hours and I seem to be able to create stuff that works well but in hindsight without being able to fathom what on earth was going through my mind at the time.  My "helpful" commentary about whatever it was I did can be similarly esoteric.

Maybe your mind does what it does whether you are looking or not.

lostmyway said:

Yeah, but at least you can write the code in the first place!

Took me long enough, though!  Nothing I've written is mathematically clever, though I do have my moments of... well, stuff that is compact and efficient, apparently.  I admit I'm guilty of the occasional instance of "drunk coding" in the early hours and I seem to be able to create stuff that works well but in hindsight without being able to fathom what on earth was going through my mind at the time.  My "helpful" commentary about whatever it was I did can be similarly esoteric.

NAS22687 said:

I've occasionally looked at the source code of some of the maths functions to see how they work and eventually just end up rather blankly staring at the screen, glassy-eyed.  Then again, I tend to do the same when reviewing code I wrote myself.

Yeah, but at least you can write the code in the first place!

Trainspotter said:

[quote][/quote]

OK, for instance, how would I work out the log of 50 to base 10?

Easy if you know what log 2 is which is .3010 (to four decimal places)

Subtract this from log 100 which makes 1.6990

(You can then work out a few other logs, such as Log 5, Log 8, Log 16, Log32, Log 25, Log 12.5, etc)

But the hard part is calculating log 2 rather than looking it up in a table .... as you need to know what power to raise 10 to make 2! Easy with a computer but very difficult and time consuming without!

I think I'm gonna have to go back to basics, Trainspotter.

I'm trying to jump into something unprepared but thanks for trying to teach me.

I've occasionally looked at the source code of some of the maths functions to see how they work and eventually just end up rather blankly staring at the screen, glassy-eyed.  Then again, I tend to do the same when reviewing code I wrote myself.

lostmyway said:

OK, for instance, how would I work out the log of 50 to base 10?

Easy if you know what log 2 is which is .3010 (to four decimal places)

Subtract this from log 100 which makes 1.6990

(You can then work out a few other logs, such as Log 5, Log 8, Log 16, Log32, Log 25, Log 12.5, etc)

But the hard part is calculating log 2 rather than looking it up in a table .... as you need to know what power to raise 10 to make 2! Easy with a computer but very difficult and time consuming without!

NAS18906 said:

I got into trouble because I took my parent's typewriter to pieces to see how it worked.

Slide rules are just a handy way of using logs and anti-logs to perform multiplication and division. They just hide the lookups that you need to do behind a non-linear scale. I bought my first slide-rule because it looked neat but I had no idea what it was for - I just wanted one of those.

Logs are all calculated by long multiplication and long division with the aid of comptometers (back in the day) and the like. As an example where you want to calculate things with decimals such as 0.17 you can by calculating the log of the number - in this case 17 and then calculating the log of one hudredth of that - 0.17 by subtracting the log of 100 from the log of 17. You only have to calculate the logs of the whole numbers and the rest is done by subtracting the logs of factors of 10.

Not sure if that makes sense but it made me think of times past!

Wish I was bright enough to understand all that, rbs.

OK, for instance, how would I work out the log of 50 to base 10?

I rather hope it does make sense: I'm hoping I'll be awake enough tomorrow to snag my concentration while it's running past in a frenzy.  But right now my brain just thinks "..." because "ooh food" is about as much of an intellectual grasp as I currently possess.

I got into trouble because I took my parent's typewriter to pieces to see how it worked.

Slide rules are just a handy way of using logs and anti-logs to perform multiplication and division. They just hide the lookups that you need to do behind a non-linear scale. I bought my first slide-rule because it looked neat but I had no idea what it was for - I just wanted one of those.

Logs are all calculated by long multiplication and long division with the aid of comptometers (back in the day) and the like. As an example where you want to calculate things with decimals such as 0.17 you can by calculating the log of the number - in this case 17 and then calculating the log of one hudredth of that - 0.17 by subtracting the log of 100 from the log of 17. You only have to calculate the logs of the whole numbers and the rest is done by subtracting the logs of factors of 10.

Not sure if that makes sense but it made me think of times past!

I had an LED TI30, a big wedge-shaped thing that ate batteries and had a glorious red display.  It didn't even need to do any calculations, it simply needed someone to enter the number 58008.618 to amuse everybody.  I think mine had eight digits (though it's been a long time) so the scope for humorous numbers was limited.

I hadn't realised that a typewriter was originally a typewriter operator.  I feel a bit ashamed of that since I'm something of a keyboard afficionado, and also remember my grandfather's typewriter, which was this really bloody enormous immovable black cast iron thing on which he'd type stuff for the BBC.  It was actually fairly smelly and unpleasant to use, but fascinating nonetheless.  And a lot less unpleasant and scary than actual typesetting: mind your ps and qs, and all that.

NAS22687 said:

Wasn't "computer" originally the term for one of the many people sitting in a big room manually working out tables with a pencil and paper?  Something like that anyway.  I could look it up but it's 6:30am, my brain isn't online yet and I'm not going to blunder around trying to find stuff until it is!

I remember seeing my late father's slide-rule and never quite figuring out how it did whatever it was that it did.  He was a draughtsman and needed to calculate stuff, and electronic calculators weren't readily available at the time (and the electromechanical ones were big, expensive and not very portable!)

A lot of terms lose their original meaning.  'Typewriter' was the name of the person who did the typing (on a typewriter!).  Secretary at that time was the person in charge of an organisation (still used today as in Secretary of State, or the General Secretary of a Club or Trade Union).  I suppose what we now know as a secretary was at that time called the 'secretary's assistant'.  No wonder we get confused!

I was quite adept with a slide rule at one time, it would certainly be a candidate for the luxury on desert island discs as it is useful, does not need batteries and it would be fun learning how to use it all over again. 

If anyone remembers when pocket electronic calculators came out in the 1970's, a lot of people had one and spent many a fascinating evening finding the square roots of numbers for nothing else but the fun of it.  Those were the days!

Trainspotter said:

The answer must be there were some very clever aspies out there at one time!

No computers, just a lot of thinking.  Of course, there was the slide rule (remember them?) which was invented around the same time as Napier was woring out logaritnmic theory.  But it is with use of advanced algebra that such was calculated.  You can get some sort of approximation by using graphs. 

I still have my slide rule, for those born before the nineteen sixties they were essential to anyone mathematical or engineering inclined.  And what is more, they worked without batteries or mains electricity!

I think you're right, Trainspotter. I'll have to to do some more studying!

I suppose using a slide rule was a more 'tactile' experience than using today's calculators, efficient though they are, and perhaps a more rewarding, if long-winded, way to do maths.

NAS22687 said:

Wasn't "computer" originally the term for one of the many people sitting in a big room manually working out tables with a pencil and paper?  Something like that anyway.  I could look it up but it's 6:30am, my brain isn't online yet and I'm not going to blunder around trying to find stuff until it is!

I remember seeing my late father's slide-rule and never quite figuring out how it did whatever it was that it did.  He was a draughtsman and needed to calculate stuff, and electronic calculators weren't readily available at the time (and the electromechanical ones were big, expensive and not very portable!)

That's correct, vometia. I saw a You Tube video the other day about it and yep, 'computer' was the name given to people who did the working out. Funny how times change.

Trainspotter said:

No computers, just a lot of thinking.  Of course, there was the slide rule (remember them?) which was invented around the same time as Napier was woring out logaritnmic theory.  But it is with use of advanced algebra that such was calculated.  You can get some sort of approximation by using graphs. 

I still have my slide rule, for those born before the nineteen sixties they were essential to anyone mathematical or engineering inclined.  And what is more, they worked without batteries or mains electricity!

Wasn't "computer" originally the term for one of the many people sitting in a big room manually working out tables with a pencil and paper?  Something like that anyway.  I could look it up but it's 6:30am, my brain isn't online yet and I'm not going to blunder around trying to find stuff until it is!

I remember seeing my late father's slide-rule and never quite figuring out how it did whatever it was that it did.  He was a draughtsman and needed to calculate stuff, and electronic calculators weren't readily available at the time (and the electromechanical ones were big, expensive and not very portable!)

lostmyway said:

I can't help wondering though how on earth people managed to work out all the log tables because you would have to find nth. roots of the base, wouldn't you?

For example, how would 10^1/8 be calculated?

The answer must be there were some very clever aspies out there at one time!

No computers, just a lot of thinking.  Of course, there was the slide rule (remember them?) which was invented around the same time as Napier was woring out logaritnmic theory.  But it is with use of advanced algebra that such was calculated.  You can get some sort of approximation by using graphs. 

I still have my slide rule, for those born before the nineteen sixties they were essential to anyone mathematical or engineering inclined.  And what is more, they worked without batteries or mains electricity!

lostmyway said:

I can't help wondering though how on earth people managed to work out all the log tables because you would have to find nth. roots of the base, wouldn't you?

For example, how would 10^1/8 be calculated?

The answer must be there were some very clever aspies out there at one time!

No computers, just a lot of thinking.  Of course, there was the slide rule (remember them?) which was invented around the same time as Napier was woring out logaritnmic theory.  But it is with use of advanced algebra that such was calculated.  You can get some sort of approximation by using graphs. 

I still have my slide rule, for those born before the nineteen sixties they were essential to anyone mathematical or engineering inclined.  And what is more, they worked without batteries or mains electricity!

Thanks, Trainspotter, that was very well explained.

I can't help wondering though how on earth people managed to work out all the log tables because you would have to find nth. roots of the base, wouldn't you?

For example, how would 10^1/8 be calculated?

The Wiki article was a bit over my head to be honest!

A logarithm is simply an index (or power).  'Natural' logarithms as used by Napier are in base 'e' (a natural number that just goes on like pi without recurring, and is  just over 2.718 ..... ). 

But a simple log table of base 2 might be easier to explain how logarithms work.

Make your own set of logarithms as follows, these are logarithms to base 2:

So

2 to the power of 1 is two

2 to the power of 2 is four

2 to the power of 3 is 8

2 to the power of 4 is 16

2 to the power of five is 32.

Now to multiple 4x8, use the powers, these are the logarithms

Therefore:

Log (base 2) of    4 is 2

Log(base 2) of     8 is 3

Log (base 2) of 16 is 4

Log (base 2) of 32 is 5

To multiply 4x8, use the logs of 4 and 8 and add them together which equals 3+2 which is equal to 5.  Then do the reverse to find the antilog (ie find out what two the power of five is by looking it up on the above table.

The antilog (base 2) of 5 is found to be 32 by looking at the table in reverse.

which is the sum we wanted to do, 4x8.

ie, 2 to the power of two multiplyed by 2 to the power of three is equal to 2 to the power of five which equals 32, which is 4x8

You could make up your own set of logarithms like this, but they would only be useful for multiplying powers of two together.  Intermediate numbers use fractional indexes (powers) which would take an awful long time to work out manually).

Hope this explains it a bit.  I read the wikipedia article and it was perhaps a little confusing if you don't understand what they are saying. The above is how I learned what logarithms are at school fifty years ago and I can remember it as clearly as anything.  We wrote the powers of two down to 2 to the power of thirty, which is a few million over 1000,000,000. (which was one thousand million when I was at school, a billion was one million million in those far off days!

Thanks, rbs, I'll study it closely.