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/
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