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Would anybody like to explain me clearly how to solve analytically this equation?

$$5.56=\frac{1-e^{-5.5x}}{1-e^{-x}}$$

I have already solved it with Mathematica and it gives $x=-0.004809$. However, I would like to know the methods for solving it analytically as a good mathematician does.

Cheers, Endora

Ng Chung Tak
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Endora
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  • I don't think this equation has a nice solution. The only thing you could do is turn this into an polynomial equation by $u^2=e^{-x}$ and look up if there are special solution techniques for the resulting polynomial. – MrYouMath Apr 11 '16 at 12:05

1 Answers1

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Let $y=e^{-0.5x}$, then

\begin{align*} 5.56 &= \frac{1-y^{11}}{1-y^{2}} \\ 5.56 (1+y) &= 1+y+y^{2}+\ldots+y^{10} \\ \end{align*}

$y$ is unsolvable in terms of radicals, though you can solve it numerically.

Ng Chung Tak
  • 18,990