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March 5th, 2013, 08:08 AM   #1
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purpose of logarithms

Could someone tell me the purpose of logarithms? I am interested in HOW the relationships between exponents and logs were developed. I can't seem to find info anywhere, I looked in my math books and get the same stuff I get the general on how to convert exponent to log, what I want to know is how this relationship was developed. So far, I get to John Napier, on wiki and found some of his work which he showed geometrically, but it was later on the relationship between log and exp was developed? Could someone either show how this came about, or point me in the right direction to a website I can check.

This would be appreciated, big thanks in advance for anyone who take the time to explain the dilemma I am having.
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March 5th, 2013, 05:21 PM   #2
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Re: purpose of logarithms

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Originally Posted by taylor_1989_2012
Could someone tell me the purpose of logarithms?
a^P = b ; logs used to calculate P given a and b...
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March 5th, 2013, 08:23 PM   #3
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Re: purpose of logarithms

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Originally Posted by taylor_1989_2012
Could someone tell me the purpose of logarithms?
There are uncountably infinitely many purposes! The initial purpose was to multiply two number easily, in a given short time. Logarithms solves the problem this way : if we want to multiply two large numbers, say, a and b, we can easily take the log of both (of course we have to look at the log chart before we do that) and add them. Now taking the exponential (depending on the base you used) will calculate the multiplication. Since I am a number theorist, I can't stop myself from saying that log (natural logs) has a great contribution in NT. It has been proved that the number of primes smaller than x is asymptotically approximately x/log(x) -- this is the beginning of analytic number theory.
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March 6th, 2013, 07:51 AM   #4
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Re: purpose of logarithms

Okay, first thanks for the replies, but I think when I wrote this, I was asking the wrong thing of sorts. Only through tedious web pages have I found my point of confusion. Here it is: I found that log are used to simplify multiplication and division problems, so I put this to the test with a simply sum do this on cal and I have the number I needed to raise get 12. But I did this on a calculator, where I want to know how they did it without. So I then found the base 10 log tables, used the values from that and found it works. Now my confusion is how do you get the values of the logs tables, how were the values worked out for base 10 in this case. I can't find information on this anywhere, I do keep getting the history of Napier, and Briggs, but no info on how they came up with the table.

Could someone please expand on this?
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March 6th, 2013, 08:57 AM   #5
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Re: purpose of logarithms

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Now my confusion is how do you get the values of the logs tables
My guess is that they used some iteration algorithm to calculate things like log(2), log(3), etc; somewhat like Newton's iteration. But there are more efficient ways to do that by generating Taylor polynomial of log(1+x). Even though it converges pretty slowly, we can accelerate it by CVZ and force it to converge it more rapidly that the original series...
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March 6th, 2013, 04:09 PM   #6
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Briggs lived some time before Taylor. Whatever method he used, it must have involved a lot of arithmetic. His tables contained a considerable number of errors.
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September 16th, 2014, 07:19 AM   #7
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The purpose of logarithm is to do fast approximate multiplications and divisions.

I took a crack at trying to understand how logs are generated. I can do it geometrically using

infinite
binary
square roots.



https://www.flickr.com/photos/859374...57641000114154
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September 16th, 2014, 07:24 AM   #8
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You don't know log until you handle a slide rule.

Virtual Pickett N909-ES SIMPLEX TRIG RULE with METRIC CONVERSION Slide Rule
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September 16th, 2014, 07:51 AM   #9
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A slide rule in the real world



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September 16th, 2014, 08:13 AM   #10
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Some reading material

Joost Burgi was the guy who developed logarithms at about the same time as Napier. He didn't really put much effort into socialising his results though, and he certainly didn't spend 20 years creating tables of millions of entries. But his version is closer to what we use today.

Basically you take an arithmetic series $a_n = n$ and a corresponding geometric series $b_n = b^n$ and pair the terms, so that $\log_b b_n = a_n$. This is all perfectly doable mathematics, but laborious.
$\{a_n\} = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \cdots \\
\{b_n\} = 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, \cdots$

Of course, you will notice that the gaps between the $b_n$ grow rather rapidly, meaning that it quickly becomes impossible to get an accurate result from your logarithms. The solution is to use a much smaller common ratio (1.0001 in Burgi's case) in the geometric series (and a large initial value $b_0 = 10^8$) to get accuracy using integer values only.

Then the table starts to take shape with $b_{n+1} = 1.0001 b_n$.

$$b_0 = 100000000 \implies \log 100000000 = 0 \\
b_1 = 100010000 \implies \log 100010000 = 10 \\
b_2 = 100020001 \implies \log 100020001 = 20\\
b_3 = 100030003 \implies \log 100030003 = 30$$
etc.

You will notice that Burgi's log values are non-standard, but they are just as effective. All that matters is the correspondence between the arithmetic sequence and the geometric sequence.

Napier's logarithms used exactly the same principal except that his logarithms $\log x$ decreased as $x$ increased. I believe he also included some changes of base when accuracy started to drop off.
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