Re: I need help. h(x) = ln(x) / x
- To: mathgroup@smc.vnet.net
- Subject: [mg12365] Re: [mg12283] I need help. h(x) = ln(x) / x
- From: Robert Pratt <rpratt@math.unc.edu>
- Date: Sun, 10 May 1998 02:04:47 -0400
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 > > >