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Solving equation involving a variable and its logarithm

  1. Aug 15, 2014 #1
    Can you suggest a general analytical solution to the following equation


    where [itex]x[/itex] is real positive variable and [itex]b[/itex] and [itex]c[/itex] are real positive
  2. jcsd
  3. Aug 15, 2014 #2


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    Gold Member

    There is no analytical solution. It should be solved numerically.
  4. Aug 15, 2014 #3

    Simon Bridge

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    Sure - it is the intersection of ##y=\ln(x)## with the line ##y=2(bx+c)/3##.
    Note: ##\ln(x)## is only defined for ##x>0##.

    Solutions will not exist for all a and b.
    In general you'll need a numerical solution.
  5. Aug 15, 2014 #4


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    You're probably right, but ... can you prove it?
  6. Aug 15, 2014 #5
    Depends on whether or not you classify the Lambert W function as an analytical solution (I doubt the mathematics community does).

    $$\ln(x^{\frac{3}{2}}) = b x + c \\
    x = \exp( \frac{2}{3}bx + \frac{2}{3}c) \\
    x \exp(- \frac{2}{3} bx) = \exp( \frac{2}{3}c) \\
    - \frac{2}{3} b x \exp(- \frac{2}{3}bx) = -\frac{2}{3} b \exp( \frac{2}{3}c) \\
    - \frac{2}{3} b x = W_n(-\frac{2}{3} b \exp( \frac{2}{3}c)) \\
    x = - \frac{3 W_n(-\frac{2}{3} b \exp( \frac{2}{3}c))}{2 b}$$

    For a real [itex]x > 0[/itex], [itex]b \neq 0[/itex] and [itex]n \in \mathbb{Z} [/itex]. Only [itex]n=-1, 0[/itex] can provide real solutions, though.
    Last edited: Aug 15, 2014
  7. Aug 15, 2014 #6


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    The W-function is certainly an analytic function so I would call that an "analytic solution".
  8. Aug 15, 2014 #7


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    Staff: Mentor

    Looks like confusion between "analytic function" and "analytic expression".
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