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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.

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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.

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