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Solving an equation with logarithms

  1. May 15, 2016 #1
    • Member warned about posting with no effort and without the template
    can somebody help me solving:

    A*ln(1+A*x)-2*ln(x)-B=0

    I thank you in advance
     
  2. jcsd
  3. May 15, 2016 #2

    Simon Bridge

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    Have you tried applying the rules for logs:
    If ln(a)=b then a=exp(b).
    ln(a*b)=ln(a)+ln(b)
    ln(a^b)=b*ln(a)
    exp(ln(a))=a
     
  4. May 15, 2016 #3

    Math_QED

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    Solving for what? What are A and B?
     
  5. May 15, 2016 #4
    for solving a physical problem (about entropy). A and B are constants

    Simon, the exponential gives a Ath degree equation.. not more solvable to me
     
  6. May 15, 2016 #5

    Math_QED

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    I get
    Well, I also get a Ath degree equation.
     
  7. May 15, 2016 #6

    Simon Bridge

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    This is why you are supposed to say what you've tried already.

    OK. You want to find an equation for x in terms of A and B?
    You got as far as:

    ##(1+Ax)^A=e^Bx^2## ... which, just at a glance, I suspect does not have an analytic solution.
    ie. taking the Ath root of both sides: ##1+Ax-e^{B/A}x^{2/A}=0##

    What was the original problem? Do you have any reason to believe the equation can be solved?
    Where does A and B come from?
    i.e. Ax<<1:
    $$x\in \frac{-A^2\pm\sqrt{A^4-4e^B}}{2e^B}$$
     
  8. May 15, 2016 #7

    Ray Vickson

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    I doubt that the equation can be solved in "closed-from", via some type of formula involving A and B. If you are given numerical values for A and B you can solve the equation numerically, using one of the many reasonably effective methods available. If ##|A|## is small we have the approximation that ##A \ln(1+Ax) \approx A^2 x##, and using that approximation in the equation leads to the (approximate) solution
    [tex] x \approx \exp \left( -W\left( -\frac{1}{2} A^2 e^{B/2} - \frac{1}{2}B \right) \right), [/tex]
    where ##W## is the so-called Lambert W-function. It is a non-elementary function: ##W(x)## is the solution of ##x = W(x) e^{W(x)}##, and is implemented in several computer algebra systems such as Maple or Mathematica.
     
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