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March 5th, 2013, 08:08 AM  #1 
Senior Member Joined: Sep 2012 Posts: 201 Thanks: 1  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. 
March 5th, 2013, 05:21 PM  #2  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,299 Thanks: 1023  Re: purpose of logarithms Quote:
 
March 5th, 2013, 08:23 PM  #3  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: purpose of logarithms Quote:
 
March 6th, 2013, 07:51 AM  #4 
Senior Member Joined: Sep 2012 Posts: 201 Thanks: 1  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? 
March 6th, 2013, 08:57 AM  #5  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: purpose of logarithms Quote:
 
March 6th, 2013, 04:09 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038 
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.

September 16th, 2014, 07:19 AM  #7 
Banned Camp Joined: Feb 2013 Posts: 224 Thanks: 6 
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 
September 16th, 2014, 07:24 AM  #8 
Banned Camp Joined: Feb 2013 Posts: 224 Thanks: 6 
You don't know log until you handle a slide rule. Virtual Pickett N909ES SIMPLEX TRIG RULE with METRIC CONVERSION Slide Rule 
September 16th, 2014, 07:51 AM  #9 
Banned Camp Joined: Feb 2013 Posts: 224 Thanks: 6  A slide rule in the real world 
September 16th, 2014, 08:13 AM  #10 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,635 Thanks: 2620 Math Focus: Mainly analysis and algebra  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 nonstandard, 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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