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Why use the natural log...?

Date: 9 Jan 1995 12:01:38 -0500
From: Gina Chen
Subject: Natural log

Dear Dr. Math,

I am a student at Monta Vista High School and I would like to know:  
Why do we have to use the natural log for the antiderivative of 1/x?  
Why doesn't the power rule work on the antiderivative of 1/x?

Thank you very much.
Gina Chen

Date: 9 Jan 1995 15:06:09 -0500
From: Dr. Ken
Subject: Re: Natural log

Hello there!

Well, let's try using the power rule on 1/x. Remember that we're looking
for a function that has 1/x as its derivative. Rewrite it as x^(-1), and
use the power rule to get the anti-derivative (x^0)/0. BAM! We'd have to
divide by zero, which we'll get in trouble for doing. Not to mention that
x^0 there, which is just a 1. So this approach won't work.

Instead, we have to define a new function to be the anti-derivative of 1/x.
Since a similar question came in recently, I'll forward that along to you.

From Ken 
Subject: Re: Natural log
Date: Sun, 8 Jan 1995 13:01:03 -0500 (EST)

Hello there!

I'll just go through and handle your questions one by one. I'm glad you're
interested in knowing more!

1) Why is e so important? Well, in a sense, e is important simply because
it has all those nice properties you've been studying. Whenever you take
the derivative of e^x (that's e to the x), you get e^x back again. It's the
only function on Earth that will do that (except things like 5 e^x and
variants like that). That's pretty cool stuff.

When I learned calculus, here's the order we defined things in: first, we
had the definite integral (from 1 to x) of 1/u du. We knew that had to be
some function of x, so we defined a new function Ln (x). It was defined as
the area under the curve 1/u. So the derivative of Ln(x) is automatically
1/x, but as of yet we hadn't looked at what this function Ln _looked_ like.

So then we used this definition of ours to figure out a few things about Ln:
we looked at Ln(ab), which was defined as the integral from 1 to ab of 1/u
du, and we decided that Ln(ab) was Ln(a) + Ln(b). "Aha!" we said. "It's
starting to look like a logarithmic function!" So then we verified that it
really was a logarithmic function, and we figured out what the base of the
logarithm was. To do this, we looked at when the function Ln(x) gave us 1.
"Whoa," we said, "that's no number I've ever seen before." Of course, we
really had seen it before, in folk tales and legends and when our big
sisters brought home their calculus homework, but this was the first time
we'd really seen it in a math class.

So we took that mysterious number and gave it a name, just in case we'd 
run into it later. As it turns out, we sure did. We ran into it in the
population growth problems, in the statistics problems, in the sequences 
and series problems, and pretty much all over the place.  So we were glad 
we gave it a name (incidentally, the "e" comes from Euler, who gave it 
its name).

Then we thought, "hey, let's turn it around. Instead of looking at the
logarithm with the base e, let's look at the exponential function to the
base e." We found that the derivative of e^x was e^x all over again.  
We learned that e^x was equal to 1 + x + x^2/2! + x^3/3! + x^4/4! + .... 
and we begged for mercy.

Or something like that. Then we learned that e^(i*Pi) + 1 = 0. This was
most impressive to us, since here was one equation that linked the five most
important numbers in mathematics: e, i, Pi, 1, and 0. It also had the three
fundamental operations: adding, multiplying, and raising to a power. And
it had the most fundamental concept in all of mathematics, that of
equality. And it had nothing else. No extra seven floating around, no
"plus c" or anything like that.  I recommend that you write it down on a
piece of paper for yourself, without all the extra junk I have to use when I
type it out on the computer, the parentheses and the caret and everything.

So that's pretty neat. What was your question again? Oh yeah. Personally,
I'd put e right on par with Pi, although some people wouldn't think so.
Certainly more people have heard of Pi; there is mention of it in the Old
Testament of the Bible, and e didn't come about until long after that
(logarithms were invented in the 16th and 17th centuries, and it probably
took a little while until people noticed that e was a nice base).

Incidentally, Logs were developed by John Napier, who lived from 1550 to
1617, and published his stuff about Logs in about 1594.  He coined the word
Logarithm, which means "number of the ratio", as in the common ratio of a
geometric sequence.  It's kind of a shame that he gave such a simple idea
such a scary name.

Anyway, e and Pi are both numbers that will pop out of your problems when
you least expect it, and I'd say that they do it with about the same
frequency.  Of course, you won't get e popping out until calculus, since you
don't define it until then (trying to define it before calculus would be
kind of hairy.  I can see it now: the teacher would say "e is a nice number
to raise to powers and to use as a base for logarithms." "Why?" "Well, I
can't tell you. Wait until calculus." They say that too much already.).

As far as there being other nice numbers that come up all the time, e and Pi
are certainly the two biggies. There's another number, called the golden
ratio, which is (1+Sqrt{5})/2. It doesn't look all that nice at first
glance, but it has some nice properties too, and the Greeks liked it a lot.
But it doesn't come up nearly as much as e or Pi, so I guess it's not on par
with the giants. I guess e and Pi are kind of the Burger King and McDonald's
of the math world, and the golden ratio is like a Hardee's or something.

So that's how I feel about e.

-Ken "Dr." Math
Associated Topics:
High School Calculus

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