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I am trying to calculate an area bordered by two functions and in the process I need to solve this equation:

$$e^{-10x}=-2x+1$$ I make it into a non-exponential form: $$-10x=ln(-2x+1)$$

And now I am stuck. Every webpage and example I have found deals with cases where all the variables are inside logarithms, but this is not the case. Can you point me in the right direction please? I know what the results should be, but I'd like to know the steps.

Thanks

Martin Melka
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  • There is a trivial solution $x=0$. The other one (which you would need for your integration problem) cannot be expressed in terms of elementary functions, but can be expressed in terms of the so-called Lambert W function. – Ian Jan 07 '16 at 18:47

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You won't find a nice algebraic answer except with the Lambert W function. Numerically you can observe that at $x=\frac12$ the left side is small and positive (about $0.0067)$ and the right is zero. You might also notice that $x=0$ is a solution. There is a root just below $\frac 12$ and you can iterate $x_{i+1}=\frac 12 (1-e^{-10x_i})$ rapidly to convergence, finding $x \approx 0.496511$

Ross Millikan
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  • I see. It's a little weird having a school problem about definite integrals that leads to this see-and-guess point, but at least I know now. Thanks! – Martin Melka Jan 07 '16 at 19:05
  • @MartinMelka Are you sure there was not some expectation that you would use software to get the integral? If not, then you probably made some setup mistake. – Ian Jan 08 '16 at 00:18
  • This is a problem a friend sent to me to have a look at. Says there was nothing about software mentioned. He submitted it with the approximation and hopefully it will be fine. For the reference, here is the task (translated: Functions f and g make up an area in R^2. Approximately draw this area and calculate its surface using definite integral.) https://fbcdn-sphotos-b-a.akamaihd.net/hphotos-ak-xpf1/v/t34.0-12/12476254_1240780935935499_1600428865_n.jpg?oh=ac96a1a0e5b25b9ee59c447aefa0f578&oe=56918EC9&gda=1452308413_4548060d52a57954da75d3d1db17ed5a – Martin Melka Jan 08 '16 at 20:21