Integration of Natural Logs

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.

Thank you for your time.

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 
you ask about,

   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/