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### Integration of Natural Logs

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Date: 02/12/98 at 17:50:50
From: Cori Jaeger
Subject: Integration of natural logs

This is a complex question so I'll state it in several ways.

Why does the natural log of x equal the integral of 1/t dt
from 1 to x?

Why is  INT[(1/t)dt] from 1 to x  the natural log of x, or
why was it defined this way?

I understand what it means, and I've seen the graph of 1/x starting
at 1 and then the integral takes the area under the curve, but
somehow I'm missing a link. I understand what the integral part
means; however, I'm not sure why it is equal to the natural log of x.

```

```
Date: 02/12/98 at 20:56:26
From: Doctor Sam
Subject: Re: Integration of natural logs

Cori,

This is a terrific question. As you suggest in your question, one
answer is "because that is how it is defined," but that leaves the
question of why it was defined that way.

You know, I assume, that y = e^x and y = ln(x) are inverse functions
of each other. That is, e^(ln(x)) = x and ln(e^x) = x. The integral
definition works because it produces a function that is the inverse
of e^x.

Here's one way to do it using implicit differentiation:

Suppose that you do know the derivative of e^x is e^x but you don't
know the derivative of y = ln x. You can create the composition
y = e^(ln x).

On the one hand we know that e^(ln x) = x so the derivative of this
function must equal 1 exactly.

On the other hand, we can differentiate using the Chain Rule to get

[e^(ln x)]' = [e^(ln x)] [ln x ]'

The derivative on the left is 1 and the bracketed expression on the
right is just x (because e^x and ln x are inverses. So we get:

1  =   x [ln x]'    Divide by x to get  [ln x]' = 1/x

Since the derivative of ln x is 1/x, the antiderivative of 1/x is
ln x.

Now you can take your original integral: INT[(1/t)dt] from 1 to x
and use the Fundamental Theorem of Calculus to evaluate the integral:

INT [ 1/t dt ]  =   ln (t) from 1 to x   =   ln x - ln 1   and since
ln 1 = 0 the result follows.

That was working backwards, from the derivative. It is also possible
to work forwards directly from the antiderivative. The integral that

L(x) =  INT [1/t dt]  from 1 to x

is an example of a function defined as an indefinite integral. Just
using properties of integrals you can show that L(x) has all the
properties of a logarithm. I don't have enough space here to go
through all those steps, but you can probably find them in most
Calculus texts if you are interested.

I hope that helps.

-Doctor Sam,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus
High School Logs

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