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natural log base (Definition)

The natural log base, or $ e$, has value

$\displaystyle 2.718281828459045\ldots $

$ e$ was extensively studied by Euler in the 1720's, but it was originally discovered by John Napier.

$ e$ is defined by

$\displaystyle \lim_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^n $

It is more effectively calculated, however, by using the Taylor series for $ f(x)=e^x$ at $ x=1$ to get the representation

$\displaystyle e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots $



"natural log base" is owned by CWoo. [ full author list (3) | owner history (3) ]
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See Also: example of Taylor polynomials for the exponential function, e is transcendental, e is irrational

Other names:  Euler number, Eulerian number, Napier's constant, e

Attachments:
e is irrational (Theorem) by mathwizard
e is not a quadratic irrational (Proof) by mathcam
$e^r$ is irrational for $r\in\mathbb{Q}\setminus\{0\}$ (Theorem) by Cosmin
$(1 + 1/n)^n$ is an increasing sequence (Theorem) by rspuzio
values of $\displaystyle \sum_{i = 0}^n \frac{1}{i!}$ for $0 < n < 26$ (Example) by PrimeFan
convergence of the sequence (1+1/n)^n (Theorem) by kfgauss70
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Cross-references: representation, Taylor series, Euler
There are 18 references to this entry.

This is version 6 of natural log base, born on 2001-11-04, modified 2008-04-29.
Object id is 657, canonical name is NaturalLogBase.
Accessed 44123 times total.

Classification:
AMS MSC: 33B99 (Special functions :: Elementary classical functions :: Miscellaneous)
Pending Errata and Addenda
None.
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Discussion
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Euler's Number by ivansayer on 2005-03-03 23:42:28
Maybe I'm thick, but I think the formula offered
should be flanked by a little explanation.
Why would you want to know about large powers
of a number just above 1 ?
The point is that if you have a list of powers
of, say, 1.00000001, there is one such number
close to any four digit decimal below ten.
If now you wish to multiply two such numbers
you take the closest powers, add the indices
and reverse lookup. This gives an *approximate*
multiplication method, and one sees that *approximate*
exponentiation and root extraction can also be carried
out. (If your result exceeds 10 you have some fancy
footwork to do, but we'll leave the reader to figure
that bit)
The only problem with this is the amount of calculation
involved and the fact that you are forever writing rather
insignificant tail digits. So ? You now divide all your
powers by n and truncate to a suitable number of digits.
This makes the number whose log was n the number whose log
is 1 - shortens your results and in no way affects their
use as approximate logs.
Of course, if you want to concatenate calculations in a
big way you have problems of accuracy and precision, but
we'll leave them for another time.

Ivan Sayer


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