Why is _e_ 'natural'?

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Neil Smith

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Jul 17, 1998, 3:00:00 AM7/17/98
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One question that has always puzzled me: when Napier was inventing
logarithms, how and why did he stumble across _e_ as the base for
natural logarithms? It doesn't seem to be the most intuitive of
choices....
----------------------------------------------------------------------
Neil Smith email: neil....@rmcs.cranfield.ac.uk
CISMG, Cranfield University, phone: +44 1793 785900
RMCS, Shrivenham, Swindon, SN6 8LA, UK fax: +44 1793 782753
Replace "nospam" with "rmcs" to reply

Matt Feinstein

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Jul 17, 1998, 3:00:00 AM7/17/98
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On Fri, 17 Jul 1998 16:06:42 +0100, Neil Smith
<neil....@nospam.cranfield.ac.uk> wrote:

>One question that has always puzzled me: when Napier was inventing
>logarithms, how and why did he stumble across _e_ as the base for
>natural logarithms? It doesn't seem to be the most intuitive of
>choices....

Perhaps the fact that ln(1+x) is approx = x for small x had something
to do with it...?

--
Matt Feinstein
mf...@aplcomm.jhuapl.edu
Organizational Department of Repeated
and Unnecessary Redundancy

Fred Schwab

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Jul 17, 1998, 3:00:00 AM7/17/98
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Neil Smith <neil....@nospam.cranfield.ac.uk> writes:

> One question that has always puzzled me: when Napier was inventing
> logarithms, how and why did he stumble across _e_ as the base for
> natural logarithms? It doesn't seem to be the most intuitive of
> choices....

> ----------------------------------------------------------------------
> Neil Smith email: neil....@rmcs.cranfield.ac.uk
> CISMG, Cranfield University, phone: +44 1793 785900
> RMCS, Shrivenham, Swindon, SN6 8LA, UK fax: +44 1793 782753
> Replace "nospam" with "rmcs" to reply


Dirk Struik, in his Concise History of Mathematics (Dover, 1967,
pp. 95-97), touches upon this question, but probably the best place
to look would be a reference he gives: F. Cajori, "History of the
exponential and logarithmic concepts", American Mathematical Monthly,
Vol. 20, 1913, 7 articles.

- Fred Schwab (fsc...@nrao.edu)
National Radio Astronomy Observatory
Charlottesville, VA, USA

Aldo Martinez

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Jul 17, 1998, 3:00:00 AM7/17/98
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Neil Smith wrote:
>
> One question that has always puzzled me: when Napier was inventing
> logarithms, how and why did he stumble across _e_ as the base for
> natural logarithms? It doesn't seem to be the most intuitive of
> choices....

I think because that number come up in nature many times. For instance
most oscillations (like ripples, pendulums, light waves, radio waves,
sound waves, etc.) are described by an equation of the form:

y(t)=A*sin(w*t)

the number _e_ comes up here because sin(x) = (e^(i*x)-e^(-i*x))/2i.
Like I said, since oscillations are very common in nature & _e_
describes them, _e_ is the 'natural' number.

Regards

Aldo

Ray Easton

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Jul 17, 1998, 3:00:00 AM7/17/98
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Neil Smith wrote:

> One question that has always puzzled me: when Napier was inventing
> logarithms, how and why did he stumble across _e_ as the base for
> natural logarithms? It doesn't seem to be the most intuitive of
> choices....

> ----------------------------------------------------------------------
> Neil Smith email: neil....@rmcs.cranfield.ac.uk
> CISMG, Cranfield University, phone: +44 1793 785900
> RMCS, Shrivenham, Swindon, SN6 8LA, UK fax: +44 1793 782753
> Replace "nospam" with "rmcs" to reply


Perhaps because if e is the base, the derivative of log(x) is 1/x. With
any other base, the derivative is more complicated. Similarly, e is the
"natural" base for exponentiation, since if this is the base, the
derivative of exp(x) is exp(x), while with any other base the derivative
is more complicated.

--
ray

--je suis marxiste, tendance groucho

Pertti Lounesto

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Jul 17, 1998, 3:00:00 AM7/17/98
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No, e is not natural, only 1,2,3,4,... are natural.
--
Pertti Lounesto http://www.hit.fi/~lounesto

mumford

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Jul 17, 1998, 3:00:00 AM7/17/98
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A while ago, Aldo Martinez <amar...@fiu.edu> begot:

>Neil Smith wrote:
>>
>> One question that has always puzzled me: when Napier was inventing
>> logarithms, how and why did he stumble across _e_ as the base for
>> natural logarithms? It doesn't seem to be the most intuitive of
>> choices....
>
>I think because that number come up in nature many times. For instance
>most oscillations (like ripples, pendulums, light waves, radio waves,
>sound waves, etc.) are described by an equation of the form:
>
>y(t)=A*sin(w*t)
>
>the number _e_ comes up here because sin(x) = (e^(i*x)-e^(-i*x))/2i.
>Like I said, since oscillations are very common in nature & _e_
>describes them, _e_ is the 'natural' number.

This is bunk...
since y(t) = A*sin(w*t) = A*(e^(i*w*t) - e^(-i*w*t))/2i
setting b = e^w, we get
y(t) = A*(b^(i*t) - b^(-i*t))/2i
b seems a much more natural base in this instance than e is, since
using b removes a term from the equation.

Back to the original poster's question...

e is the base of the natural logarithm for the same reason that radians
are used instead of degrees... it reduces the math to its simplest:

d(ln(x))/dx = 1/x
while
d(log_b(x))/dx = d(ln(x)/ln(b))/dx = 1/(x*ln(b)) = (log_b e)/x

One of the ways that it is argued logarithms were invented was
to start out with a statement p(xy) = p(x) + p(y), and from there
to evaluate derivatives and calculate inverses. The derivative
comes out to p'(x) = p(e)/x, which is obviously more simple if
you chose a p such that p(e) = 1.

Also, I was not aware that Napier 'invented' logarithms. As far
as I knew, he just built a (now famous) table of them, and in base
10 as well.

--
Glenn Lamb - mum...@netcom.com. Finger for my PGP Key.
Email to me must have my address in either the To: or Cc: field. All other
mail will be bounced automatically as spam.
PGPprint = E3 0F DE CC 94 72 D1 1A 2D 2E A9 08 6B A0 CD 82

Mark Rajesh Das

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Jul 17, 1998, 3:00:00 AM7/17/98
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Pertti Lounesto (loun...@maanantai.hit.fi) wrote:
: No, e is not natural, only 1,2,3,4,... are natural.
I don't think he/she was saying that e is a natural number, but was
speaking in reference to the "natural logarithm"

Which, on an aside, reminds me of something somewhat funny that my
analysis Prof said. He said that we use log base 10 because we have
ten fingers and then said that nature uses base e which means God must
have 'e' fingers.

It was funnier during lecture I guess.

: --
: Pertti Lounesto http://www.hit.fi/~lounesto

Kurt Foster

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Jul 17, 1998, 3:00:00 AM7/17/98
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In <35AF68...@nospam.cranfield.ac.uk>, Neil Smith said:
. One question that has always puzzled me: when Napier was inventing
. logarithms, how and why did he stumble across _e_ as the base for
. natural logarithms? It doesn't seem to be the most intuitive of
. choices....
.
He didn't - not quite. According to "A History of Mathematics" by
Boyer, what Napier DID do was to define, for a given number N, a
"logarithm" (a word he coined) L by the equation

N = 10^7 * (1 - 1/10^7)^L.

This L is very nearly equal to -ln(N/10^7). Napier's geometrical
considerations leading to this definition are also described.
I'm not sure of the etymology of the description "natural", but the
naturalness of "e" as a base for exponential and logarithmic functions
is evident from the limits

exp(x) = LIMIT, h -> 0, (1 + h*x)^(1/h) [exponential to base "e"]

ln(x) = LIMIT, h -> 0, (x^h - 1)/h (x > 0) [log to base "e"]

and the differential equations

d/dx(exp(x)) = exp(x)

d/dx(ln(x)) = 1/x.

If you use a different base "b" for your log or exponential functions,
then nasty numerical factors of ln(b) start creeping into derivatives and
integrals involving logarithmic or exponential functions to the base b.
The notation "e" for the number exp(1) = 2.71828+ is due to Euler.

Yves Gallot

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Jul 17, 1998, 3:00:00 AM7/17/98
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His definition of e is:
Integral[1, e] dx/x = 1.
Considering the function 1/x is more natural than any other a/x.

Yves


Mike McCarty

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Jul 18, 1998, 3:00:00 AM7/18/98
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It is natural in exactly one sense, it is the one which has a "simple"
method for computation.

ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 ...

(actually, if you want to use a series, it's better to use other series)

Mike

In article <35AF68...@nospam.cranfield.ac.uk>,
Neil Smith <neil....@nospam.cranfield.ac.uk> wrote:
)One question that has always puzzled me: when Napier was inventing
)logarithms, how and why did he stumble across _e_ as the base for
)natural logarithms? It doesn't seem to be the most intuitive of
)choices....
)----------------------------------------------------------------------
)Neil Smith email: neil....@rmcs.cranfield.ac.uk
)CISMG, Cranfield University, phone: +44 1793 785900
)RMCS, Shrivenham, Swindon, SN6 8LA, UK fax: +44 1793 782753
) Replace "nospam" with "rmcs" to reply


--
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This message made from 100% recycled bits.
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a_st...@my-dejanews.com

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Jul 18, 1998, 3:00:00 AM7/18/98
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In article <35AF74...@fiu.edu>,
amar...@fiu.edu wrote:

> Neil Smith wrote:
> >
> > One question that has always puzzled me: when Napier was inventing
> > logarithms, how and why did he stumble across _e_ as the base for
> > natural logarithms? It doesn't seem to be the most intuitive of
> > choices....
>
I think this came up from the "natural" correspondence there is between an
arithmetic and a geometric progression . if x is a positive reall number, then
we have a on-to-one correspondence betwen the following A.P. and G. P.

....... -2x -x 0 x 2x........
.....(1+x)^(-2) (1+x)^(-1) 1 (1+x) (1+x)^2......

From this we can create a base foe a logarithm system. It's clear the if b is
such base, then b = (1+x)^(1/x).

Napier probably found it "natural" that someone would try to make x as close
to zero as possible, so that the set of numbers belonging generated by the AP
and the PG became dense, that is, all of their elements were cluster points.
By doing this you "naturally" build a continuous logarithim system. And then
e arises as e = lim (1+x)^(1/x) =~ 2.718281828...... x=>0

Do you agree this is a "natural" process?
Artur Steiner

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V. Rezzonico

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Jul 18, 1998, 3:00:00 AM7/18/98
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Neil Smith wrote:
>
> One question that has always puzzled me: when Napier was inventing
> logarithms, how and why did he stumble across _e_ as the base for
> natural logarithms? It doesn't seem to be the most intuitive of
> choices....

If we ever met some aliens, the numbers we'll have in common would be pi
and e, because in every natural law of increment there's an e somewhere.

Then e^x is the only function that when derivated is still the same.

Moshe Zadka

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Jul 18, 1998, 3:00:00 AM7/18/98
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On Sat, 18 Jul 1998, V. Rezzonico wrote:

> If we ever met some aliens, the numbers we'll have in common would be pi
> and e, because in every natural law of increment there's an e somewhere.
>
> Then e^x is the only function that when derivated is still the same.

Wrong, but close.
The only solutions to d/dx(f)=f are the functions
a(e^x)

The only solutions to d/dx(f)=f and (boundary condition, normalization
assumption) f(0)=1
is e^x.

Constantly nitpicking...
--
Moshe Zadka - moshez (at) math (dot) huji (dot) ac (dot) il
What's Yellow and Complete? A Bananach Space


Zdislav V. Kovarik

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Jul 18, 1998, 3:00:00 AM7/18/98
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In article <35B055...@trevano.ch>,
V. Rezzonico <rezz...@trevano.ch> wrote:

:Neil Smith wrote:
:>
:> One question that has always puzzled me: when Napier was inventing
:> logarithms, how and why did he stumble across _e_ as the base for
:> natural logarithms? It doesn't seem to be the most intuitive of
:> choices....
:
:If we ever met some aliens, the numbers we'll have in common would be pi

:and e, because in every natural law of increment there's an e somewhere.
:
:Then e^x is the only function that when derivated is still the same.

Not again. Just differentiate (by x) 73*e^x, and then correct yourself.
(If you require that the value at 0 be 1, you get somewhere.)

Cheers, ZVK(Slavek).

David Kastrup

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Jul 18, 1998, 3:00:00 AM7/18/98
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Neil Smith <neil....@nospam.cranfield.ac.uk> writes:

> One question that has always puzzled me: when Napier was inventing
> logarithms, how and why did he stumble across _e_ as the base for
> natural logarithms? It doesn't seem to be the most intuitive of
> choices....

e^x = lim(n->oo) (1+x/n)^n

Which means that if you take discrete geometric growth problems and
let the intervals as well as the growth factor shrink at the same
time, you arrive at a relation including e.

If I take some capital and get 100/365 % of interest per day for 365
days, I'll have pretty much e times that capital after one year.


--
David Kastrup Phone: +49-234-700-5570
Email: d...@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
Institut für Neuroinformatik, Universitätsstr. 150, 44780 Bochum, Germany

Allen Adler

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Jul 18, 1998, 3:00:00 AM7/18/98
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In reply to Neil Smith's question:

> One question that has always puzzled me: when Napier was inventing
> logarithms, how and why did he stumble across _e_ as the base for
> natural logarithms? It doesn't seem to be the most intuitive of
> choices....

various posters have given reasonable explanations of why e can
be regarded as natural. However, strictly speaking, Neil Smith's
question pertains to what Napier's actual line of thought was.
For this, it would be good to have a look at Napier.

Question: Where, in a more or less modern edition, can one read what
Napier actually wrote about this? For example, is there an
edition of Napier's collected works, so that while looking
this up one can find out about other mathematics he did.

Allan Adler
ad...@hera.wku.edu

Mark-Jason Dominus

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Jul 18, 1998, 3:00:00 AM7/18/98
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This article by Matthew P Wiener sheds a lot of light on the
appearance of e.

From: wee...@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: Why are Natural logs?
Date: 22 Dec 1997 17:31:12 GMT
Message-ID: <67m850$q3d$3...@netnews.upenn.edu>

In article <67jg1i$c...@freenet-news.carleton.ca>, do440@FreeNet (Orin Durey) writes:
>What is it that determines the value of e in a natural log?

>From two earlier USENET articles of mine on e:
========================================================================
For a detailed account, see Eli Maor E: THE STORY OF A NUMBER (Princeton,
1994).

>You define a^x and log_a for any positive a. Trying to evaluate the
>derivative of these functions, you immediately notice, that you need
>only the derivative of log_a at 1. This is (if it exists, which all
>historical men had no doubt about) lim_{n to infty} n log_a(1+1/n) =
>lim log_a (1+1/n)^n. You are automaticall led to the limit of
>(1+1/n)^n.

This is true as mathematics, but revisionist as history.

Napier, who invented logarithms, more or less worked out a table of
logarithms to base 1/e, as follows:

0 1 2 3 4 5 6 7 8 9 10 ...
1 2 4 8 16 32 64 128 256 512 1024 ...

The arithmetic progression in the first row is matched by a geometric
progression in the second row. If, by any luck, you happen to wish to
multiply 16 by 32, that just happen to be in the bottom row, you can
look up their "logs" in the first row and add 4+5 to get 9 and then
conclude 16*32=512.

For most practical purposes, this is useless. Napier realized that what
one needs to multiply in general is 1+epsilon for a base--the intermediate
values will be much more extensive. For example, with base 1.01, we get:

0 1.00 1 1.01 2 1.02 3 1.03 4 1.04 5 1.05
6 1.06 7 1.07 8 1.08 9 1.09 10 1.10 11 1.12
12 1.13 13 1.14 14 1.15 15 1.16 16 1.17 17 1.18
18 1.20 19 1.21 20 1.22 21 1.23 22 1.24 23 1.26
24 1.27 25 1.28 26 1.30 27 1.31 28 1.32 29 1.33
30 1.35 31 1.36 32 1.37 33 1.39 34 1.40 35 1.42
[...]
50 1.64 51 1.66 52 1.68 53 1.69 54 1.71 55 1.73
[...]
94 2.55 95 2.57 96 2.60 97 2.63 98 2.65 99 2.68
100 2.70 101 2.73 102 2.76 103 2.79 104 2.81 105 2.84
[...]

So if you need to multiply 1.27 by 1.33, say, just look up their logs,
in this case, 24 and 29, add them, and get 53, so 1.27*1.33=1.69. For
two/three digit arithmetic, the table only needs entries up to 9.99.

Note that e is almost there, as the antilogarithm of 100. The natural
logarithm of a number can be read off from the above table, as just 1/100
the corresponding exponent.

What Napier actually did was work with base .9999999. He spent 20 years
computing powers of .9999999 by hand, producing a grand version of the
above. That's it. No deep understanding of anything, no calculus, and
e pops up anyway--in Napier's case, 1/e was the 10 millionth entry. (To
be pedantic, Napier did not actually use decimal points, that being a new
fangled notion at the time.)

Later, in his historic meeting with Briggs, two changes were made. A
switch to a base > 1 was made, so that logarithms would scale in the
same direction as the numbers, and the spacing on the logarithm sides
was chosen so that log(10)=1. These two changes were, in effect, just
division by -log_e(10).

In other words, e made its first appearance rather implicitly.

The calculus connection came later. Fermat had successfully solved the
quadrature problem for y=x^n for n!=-1, but not for y=1/x. Fermat's
method was to use geometrically spaced intervals on the x axis, and to
add the resulting areas. It took a bit of time for a contemporary to
notice that this method produced arithmetically spaced areas under the
hyperbola--ie, that there's a logarithm going on.
========================================================================
>To Matthew Wiener's interesting post on the history of logarithms, I
>add only this footnote: before Napier invented logarithms, tables of
>trig functions were used to convert multiplication problems to
>addition, subtraction, and division by two, via the formulas for
>sin(A)sin(B) in terms of sin(A+B) and sin(A-B).

Oy, I forgot about that!

Technically speaking, sin(A)sin(B) was expressed directly in terms of
cos(A+B) and cos(A-B), as was cos(A)cos(B). Mixed products cos()sin()
were expressed in terms of sin(A+B) and sin(A-B). Of course, you can
convert one to the other by complementary angles, but I had a momentary
jar over what you said.

Anyway, anyone still hungering to get a ground feel for `e' and natural
logarithms, below is a C program you can use. Call it nap.c, and compile
it via "cc nap.c -lm -o nap" and run it as "nap 1.01" or "nap 1.001" or
the like, at least on Unix systems. The output for nap 1.001, starts:

n b^n n b^n n b^n n b^n n b^n
0.000 0 1.000 1 1.001 2 1.002 3 1.003 4 1.004
0.005 5 1.005 6 1.006 7 1.007 8 1.008 9 1.009
0.010 10 1.010 11 1.011 12 1.012 13 1.013 14 1.014
0.015 15 1.015 16 1.016 17 1.017 18 1.018 19 1.019
0.020 20 1.020 21 1.021 22 1.022 23 1.023 24 1.024

The first column is log (base e) of the third column, which in theory is
close to n(b-1) for b-1 small. So if you want to multipy 123 * 456, you
can look up 1.23 and 4.56 in the powers side:

0.205 205 1.227 206 1.229 207 1.230 208 1.231 209 1.232
[...]
1.514 1515 4.546 1516 4.551 1517 4.555 1518 4.560 1519 4.564

giving 123=100*b^207 and 456=100*b^1516, so 123*456=10000*b^(207+1516)=
10000*b^1723=55960, actual value 56088. So it's good to about 3 digits.

Where does `e' come in here? When n=1000, we have:

1.000 1000 2.717 1001 2.720 1002 2.722 1003 2.725 1004 2.728

so b^1000 is about e. This is in accordance with (1+1/M)^M -> e. The
above can thus be rewritten as 123=100*e^.207 and 456=100*e^1.516. The
reason why switching to `e' is so "natural" is that the choice of b is
rather arbitrary, and normalizing the exponents allows one to compare
1.01^21 with 1.001^207 with 1.0001^2072 without having to correct for the
scale each time. (One can normalize anywhere, actually, but obviously
the bottom barrel simple normalization is at e.)

This wasn't noticed for some time though. Presumably the effort involved
in calculating log tables was so great that the rather obvious tendencies
visible above had to wait until theory predicted it.

And in conclusion, let me again remind you how the table ends at 10.0:

2.299 2300 9.963 2301 9.973 2302 9.983 2303 9.993

In short, at about log_e(10)=2.30258.... As we all know, log_e(X)/log_e(10)
equals log_10(X), but Napier and Briggs did not know that. Rather, they
were doing something much more practical. When it comes to multiplying
numbers written out in decimal notation, multiples of 10 can be factored
out, so it was more natural for their purposes to have the table wrap at
10, which they insured by rescaling by whatever gave 10.000 in the end.
Since their only intent was multiplication, the actual scaling of the
logarithms mattered not.

I consider it a real shame that none of this is taught any more, one of the
reasons I'm willing to write at length on such a "trivial" topic. It may be
mathematical convenience to define log via an integral, but today's students
are being cheated and snowed over. It *is* possible to proceed by defining
exponentials and logarithms first, and then if one works out logarithms in
the spirit of Napier (with a little calculating help), one sees *visibly*
all their basic properties, including (1+1/M)^M -> e.
------------------------------------------------------------------------
#include<stdio.h>
#include<math.h>

main(int argc,char **argv)
{
int i;
double x,b;

if(argc!=2)
{
printf("Usage: %s base\n",*argv);
exit(1);
}

sscanf(argv[1],"%lf",&b);

if(b<=1.)
{
printf("Base must be > 1.0\n");
exit(2);
}

printf(" "); /* headers */
for(i=0; i<5; i++)
printf(" n b^n ");

for(i=0,x=1.; x<10.0; i++,x*=b)
{
if(i%5)
printf(" ");
else /* first column is natural logarithm */
printf("\n%7.3f",log(x));
printf("%6d%7.3f",i,x);
}

printf("\n");
exit(0);
}
------------------------------------------------------------------------
========================================================================
--
-Matthew P Wiener (wee...@sagi.wistar.upenn.edu)

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