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Re: I need help. h(x) = ln(x) / x



You are trying to find intersection points of f(x)=n^x and g(x)=x^n. Set
them equal:
				n^x = x^n
Now take ln on both sides:
				ln(n^x) = ln(x^n)
				x ln(n) = n ln(x)

Divide both sides by n x:
				ln(n)/n = ln(x)/x

That is,
				h(n) = h(x)

You will find that h has a maximum at (e, 1/e) and a (right) horizontal
asymptote y=0.

Solutions to the original equation correspond to x values x1=n and x2=x
with the same y value on the curve y=h(x).  For example, take x1=2 and
x2=4.  h(2) = h(4), so 2^4 = 4^2.  By the way, this is the only
nontrivial (n and x unequal) solution with n and x both integers, as
you can verify by analysis of the function h.

Problems related to the equation n^x = x^n have appeared on the Putnam
exam a few times.

I hope I didn't answer too much.

Rob Pratt
Department of Mathematics
The University of North Carolina at Chapel Hill CB# 3250, 331 Phillips
Hall
Chapel Hill, NC  27599-3250

rpratt@math.unc.edu

http://www.math.unc.edu/Grads/rpratt/

On Thu, 7 May 1998, Michael wrote:

> Hello,
> I need some help. I need to know the link between h(x) = ln(x) / x and
> (f(x) = n^x and g(x) = x^n). So I don't want you do the below question.
> (because I can do that).  I just need to know the Link between them. Oh
> ln(x) means the loge(x) / x.
> 
> ------------------------------------------------------------------------------------------------------------------------------
> 
> PLEASE DON'T DO THE BELOW!!!! (Look at the question above) To complete
> her investigation into the problem Jan now decides to take x to be a
> positive real number.  Sketch the graph of the function h(x) = ln(x) /
> x over an appropriate domain.  Discuss key features of the graph. For n
> > 2, use this graph to investigate the number of intersection points
> and regions of the domain within which these points of intersection
> occur, for pairs of functions of the form f(x) = n^x and g(x) = x^n.
> 
> Michael
> P.s. if you would it out you have SAVED my life. 10 times over
> 
> 
> 




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