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table of natural logarithms

Major Section: 
Reference
Type of Math Object: 
Data Structure
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Mathematics Subject Classification

26-00 no label found26A09 no label found26A06 no label found

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So as I understand the concept, logarithms used to help perform arithmetic with large numbers that is so often crucial to science. I also understand that logarithms "step down" multiplication to addition, division to subtraction, etc. But what I don't understand is: how did they step the result back up? Even with base 10 logarithms, the result would often be a number with a fractional part, so getting the result wouldn't be as easy as writing a bunch of zeroes after a one. So something is missing in this picture.

Well, next time you yell at your computer for taking a little long with an operation, be grateful you have a computer at all.

Is your question "why aren't their exponential tables?" That is, if you wanted to go back and that required that you find 10^1.23, how would that be done before computers?

Well, log tables also are exponential tables because logs are bijections. You just read the table differently. To find 10^1.23 you in a log table means that you look in a log base 10 table and find the box which has 1.23 (or the closest to that). Then you pullback to find the number which represents 10^1.23.

This same sort of thing happens with many data tables, such as tables of P-values in statistics. You don't bother writting two tables with the same data but just assume that the reader understand the meaning of the data well enough to use it forward and backward.

Put even more simply: you use the table of logaritmhs to step down the inputs and you use the same table to step up the output.

Provided of course that the table goes far enough in the necessary direction. Here's an example contrived not to be out of range for this table: multiply 12 by 8. Log 12 is 2.48491, according to the table, and log 8 is 2.07944. Add those logs up you get 4.56435. You look 4.56435 up in the table and find it at the position corresponding to 96. Obviously it would've been easier to just mentally calculate 8 * 10 + 8 * 2, which is why Babbage's table went up to 108,000. However, I think Babbage's table only had four decimal places for each entry, so in that sense this table is "wider" than Babbage's.

There is something almost miraculous about actual numbers. A scary concept like negative infinity (for log 0) doesn't seem as foreboding put in the company of numbers like 2.3 and 0.6.

> But what I don't understand is: how did they step the result back up?

They did that by looking a the logarithm table backwards or, if they had
an antilogarithm table, by looking at that. Let me illustrate with an
example. Suppose we want to multiply 17.81 by 6.520. Whipping out
the old log table (and blowing the dust off it nowadays), I find that
the base 10 logarithms of these numbers are 1.25066 and 0.81425.
Adding them produces 2.06491. To step the result back up, I look in
the table for the logarithm closest to 0.06491 and find that
log 1.161 = 0.06521 so the answer is 116.1 to four digits.

> Even with base 10 logarithms, the result would often be a number with
> a fractional part, so getting the result wouldn't be as easy as
> writing a bunch of zeroes after a one.

As shown in the example above, the fractional part would be dealt with
by looking in the table backwards to find a number between 1 and
10 whose logarithm is closest to the fractional part of the result.
Then obtaining the result is a matter of taking into account the
integer part by adding zeros and shifting the decimal point
appropriately.

> So something is missing in this picture.

It would be good if that something were explained in the entry on
logarithm tables. Along those lines, I would suggest the following
improvements. Change the title to "logarithm tables", then add
a short table of base 10 logarithms and add some text explaining
how log tables were used to multiply and divide numbers in the
old days. While this may not be of much contemporary relevance, it
is a part of the recent history of mathematics, so it definitely
is worth describing in order to round out our account of math.

There is already an entry for logarithmic calculation
http://planetmath.org/encyclopedia/BriggsianLogarithms.html
There one does not calculate a product of numbers but a third root, i.e. a power with exponent 1/3. I also illustrate how the place of decimal point in the numerus affects on the integer part ("characteristica") of the logarithm of the numerus.
Regards,
Jussi

rspuzio writes:
"It would be good if that something were explained in the entry on
logarithm tables. Along those lines, I would suggest the following
improvements. Change the title to "logarithm tables", then add
a short table of base 10 logarithms and add some text explaining
how log tables were used to multiply and divide numbers in the
old days. While this may not be of much contemporary relevance, it
is a part of the recent history of mathematics, so it definitely
is worth describing in order to round out our account of math."

I couldn't have put it better myself. In fact, I wasn't able to put it at all earlier. I remember Ray's excellent entry "How to use a table," and something along those lines is called for here. I can't do it today, though, I have some mundane matters to attend to which will distract me for the rest of today. Perhaps tomorrow I can come back to this, but in the interim I won't mind if anyone else wants to address this deficiency.

Thanks to everyone who took the time to explain this to me. I used to nod in agreement to the saying "God made the integers and man made the rest." But logarithms give me an appreciation that there is also a sort of divine order to the real numbers as well. Thanks again, everyone!

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