1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proving e^x=x^e for only one positive x

  1. Jul 6, 2015 #1
    Hi, I'm teaching myself calc 2, and could use some help with a problem I'm not sure how to solve:

    Problem:
    Prove that x^e=e^x has only one positive solution.

    Relevant equations (I think?):

    b^x = a^(loga(b)*x) - base change for logarithms

    The attempt at a solution:

    e^x=x^e=e^(ln(x)*e) - base change
    x = ln(x)*e - powers of equal bases
    At this point I'm stuck, I'm not sure how to show that x = ln(x)*e has only one positive solution.
     
    Last edited by a moderator: Jul 6, 2015
  2. jcsd
  3. Jul 6, 2015 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Try to argue for what happens when x < e and when x > e, respectively. (Obviously, x = e is a solution.)
     
  4. Jul 6, 2015 #3
    I tried assuming x=e+a, and get e^e+a=(e+a)e=eln(e+a)*e, which gives (e+a)/e=ln(e+a), or e(e+a)/e=e+a, but I haven't been able to reach a clear absurdity.
     
  5. Jul 6, 2015 #4

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Try to use inequalities rather than working with a constant.
     
  6. Jul 17, 2015 #5
    Try plotting graph of x∧(1/x) for x>0 .

    You will see that it is increasing for x<e and decreasing for x>e .

    Now compare x∧(1/x) with e∧(1/e) . The same value is only possible for x=e .

    Raise both to x*e and you will get your answer .
     
  7. Jul 17, 2015 #6
    I always start these deals with graphing.

    Once you graph it and see that the functions are increasing at different rates, it will become clear how to prove they only intersect once.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Proving e^x=x^e for only one positive x
  1. Integral of x(e^x) (Replies: 1)

  2. E^x > f(x) (Replies: 1)

Loading...