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X*a^x=b, how to solve this?

  1. May 1, 2006 #1
    x*a^x=b

    how to solve this?
     
  2. jcsd
  3. May 1, 2006 #2

    Curious3141

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    Homework Helper

    You can't, in general, solve it exactly. Use a numerical method to approximate the roots.
     
  4. May 1, 2006 #3

    VietDao29

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    Homework Helper

    This is a good candidate for Newton's method.
    You first choose an x0 arbitrarily. You can graph it first, then choose an x0 wisely so it's near the roots.
    Then use the formula:
    [tex]x_{n + 1} = x_n - \frac{f(x_n)}{f'(x_n)}[/tex], and let n increase without bound to obtain the solution, i.e:
    [tex]x = \lim_{n \rightarrow \infty} x_n[/tex].
    Can you get this? :)
     
  5. May 1, 2006 #4
    Yes.
    And it works! :)
    I didn't know about Newton's method before.
     
  6. May 3, 2006 #5
    with math reasoning

    Suppose a and b are unknown constants
    Let [tex]y=xa^x[/tex] and hence [tex]y=b[/tex]
    Take a ln of bothe sides leading to [tex]lny=lnx+xlna[/tex]
    We understand that x,a,b must be > 0

    Taking a derivative of y gives us
    [tex]\frac{dy}{dx}=(\frac{1}{x}+lna)xa^x[/tex]
    Now we find that [tex]x=0 (omitted), \frac{-1}{lna}[/tex]

    Next, we draw a table to check signs of [tex]\frac{dy}{dx}[/tex], but before that we check [tex]a[/tex]
    1. if 0<a<1
    Look at the table and mark for sign (+/-), then check for y to compare with [tex]y=b (a.straight.line)[/tex], which means you need to reason the value of b for where the root(s) exist.
    2. if a>1
    Do the same to find out root domain

    Now things become easier when you know concrete constant a, b. just put them inthere to find a root. This way looks crary though :biggrin: but sovable domain can be understood
     
    Last edited: May 3, 2006
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