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I'm trying to solve a bigger problem however I am stuck at this step:

How can I solve:

$$ 2^x - x = 5 $$

any hints/tips/steps please?

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    Trial and error? 1 doesn't work, 2 doesn't work, ... (If you plot $y=2^x$ and $y=5+x$ in the same diagram, you'll see that there are two solutions, but I don't think the second one has a simple closed form.)2011-10-23
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    Normally I might say something about Newton's method or about attractive fixed points. But in this case the answer is staring you in the face.2011-10-23
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    One solution is "obvious". The other real solution needs the services of Lambert.2011-10-23
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    the second solution is somewhere between $-5$ and $-4$2011-10-23
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    $\approx -4.969$2011-10-23
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    Thanks, I know the answer is obvious in this case and can be found via trial and error. I was just wondering if there is a more concrete method that could be used for cases when the answer is not obvious.2011-10-23

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As alluded to in the comments there is an integer solution. For the other solution is existence of a solution good enough? You can use the Intermediate Value Theorem on the function $f(x)=2^x-x-5$. It is negative at $x=0$ and positive at $x=-6$. So, somewhere in between the IVT says there must be a $0$. Or you can use Newton's method on $f$ to approximate the $0$ of $f$.