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Thread: advanced transposition problem

  1. #1
    May 2008

    Question advanced transposition problem

    I'm having massive difficulty in rearranging the following to find d


    taking the natural log of both sides normally helps when there is an exp function present but here it just gets in the way - no matter how hard I try I can't get d on its own (d in terms of c and r).

    can anyone help?

    showing the steps you took would help me tackle similar problems in the future so I'd be extremely grateful for this.

    Massive thanks in advance.

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  2. #2
    MHF Contributor
    Aug 2007
    There is no Algebra I method to solve such things.

    Have a look at Lambert's W-Function. This will suggest to you various methods that have been contemplated to deal with such structures.

    Lambert W-Function -- from Wolfram MathWorld
    Lambert W function - Wikipedia, the free encyclopedia
    Lambert's W-Function

    On the other hand, if you do not already know of the difficulty of solving what you attempted to solve, these discussions may not suggest much that you will find useful.

    On still another hand, have you considered what happens as 'd' increases without bound? $\displaystyle \frac{c}{h} = -2r^{2}$ Does that lead to anything useful?

    On yet another hand, after substituting $\displaystyle q = \frac{1}{d^{2}}$ and rearranging, we have $\displaystyle \frac{cq}{h} = 1-e^{2qr^{2}}$

    ...and we can contemplate what happens for q = 0.
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  3. #3
    May 2008

    Thumbs up Thanks

    Many thanks for such a comprehensive reply. It's beyond the scope of my current level of understanding but i'll endevour to fill in the blanks by doing some background reading.

    Thanks once again

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