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I'm studying properties of logarithm but I don't understand how base e works. Base 10 looks simple while doing calculations of numbers having multiple of 10. As other numbers are not multiple of 10 how one can calculate without using log table. That's why I want know how log table is constructed & how base e works ?
I'm very basic user, I don't get integration, derivatives & summations. Please give answer theoretically or using row concepts. Since I want to construct log table by myself.

Vikas.Ghode
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    I don't think this question is too broad. However, it needs a careful answer. I'll look at this tonight, if nobody has given an answer. – Jean-Claude Arbaut Oct 06 '15 at 08:57
  • Essentially you want to know about the number $e$. 1:http://math.stackexchange.com/questions/407286/two-questions-about-eulers-number-e 2:http://math.stackexchange.com/questions/784022/defining-the-number-e 3:http://math.stackexchange.com/questions/707570/what-are-the-uses-of-eulers-number-e 4:http://math.stackexchange.com/questions/3319/why-is-the-number-e-so-important-in-mathematics – Aditya Agarwal Oct 06 '15 at 09:25
  • Is there any answer that doesn't contains integration, derivative, summation ? That's why no link is useful to me by @aditya agrawal – Vikas.Ghode Oct 06 '15 at 10:06
  • Logarithms were invented by Napier in 1614, almost 30 years before Newton's birth. So yes, basically, you can develop logarithms without analysis though it will be much more cumbersome. Analysis gives you more power to explain how they work and how to compute them efficiently, using quickly converging series instead of repeated square roots, for instance. – Jean-Claude Arbaut Oct 06 '15 at 10:35
  • See representations section in https://en.wikipedia.org/wiki/E_(mathematical_constant) – Aditya Agarwal Oct 06 '15 at 10:40
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    Maybe of interest: http://math.stackexchange.com/questions/698384/approximating-logs-and-antilogs-by-hand – Jean-Claude Arbaut Oct 06 '15 at 13:58
  • @jean-claude atbat it helped me alot. &raised new problems about antilogs. I also wanted to do all calculations in mind. Thanks. – Vikas.Ghode Oct 06 '15 at 14:58

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"Logarithm" is the inverse to "exponential". In particular, if $y= a^x$, then $log_a(y)= x$. For example, it is true that $10^{0.30102}= 1.9999$ (approximately 2) so log(2) is approximately log(1.9999)= .30102. If you really want to construct a table of logarithms, base 10, using paper and pencil, as mathematicians several hundred years ago, be prepared to do some tedious arithmetic (as they did)! log(0.01) is easy- 0.01= 1/100= $10^{-2}$ so log(0.01)= -2. log(0.02) is harder. You can simplify a bit by arguing that log(0.02)= log(2/100)= log(2)- log(100)= log(2)- 2 so you just need to find log(2) (in fact, many tables of logarithms just give logarithms of numbers from 1 to 10). log(2) is the x such that $10^x= 2$. If we did not know that $10^{0.30102}$ is approximately 2, we could work it out by noting that $10^{0.3}= (10^3)^{1/10}= 1000^{1/10}$ and start calculating that value, then noting that $10^{0.4}= 10^{2/5}= 100^{1/5}$ and seeing that the first is less than 2 while the second is larger than 2- so the logarithm is between .3 and .4. So look at $10^{3.5}$. As I said, be prepared for so long tedious calculation.

user247327
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