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How to solve this equation?

$$ x = 10^{x/10} $$

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There is an obvious solution x = 10 and the RHS grows faster than the LHS. – Qiaochu Yuan Aug 3 '10 at 2:50
@Qiaochu Yuan: Even if the answer is trivial and only takes seconds, I think you should post it as an answer, so that the question can accept an answer and reach "closure". :-) – ShreevatsaR Aug 3 '10 at 3:06
Besides, that's not a complete answer. x ≈ 1.37129 works as well. – Ben Alpert Aug 3 '10 at 3:18
@Kaestur To solve an equation means to find all its solution and to prove that there are no other solutions. – Grigory M Aug 3 '10 at 6:45
@Kaestur probably we should give OP the benefit of the doubt for now, then – Grigory M Aug 3 '10 at 8:01
up vote 20 down vote accepted

There is an obvious solution $x = 10$. For $x > 10$ the derivative of the RHS is at least $\log 10 > 1$ so there are no solutions. For $x \le 0$ there are obviously no solutions. By the IVT there is a solution in $(0, 10)$, and by convexity this solution is unique. In fact this solution is in $(1, 2)$. It can be expressed using the Lambert W-function, but it is really not worth writing down explicitly. Numerically it is about $1.37$.

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This solves the problem - but does not answer the question. The question was not 'solve this' but rather 'how'. One person's obvious is another's mystery. – Jacques Carette Aug 4 '10 at 0:06
I think the methods largely speak for themselves. If the OP has a question about them he/she should ask in a comment and I will be glad to clarify. – Qiaochu Yuan Aug 4 '10 at 3:13

You can study and graph the two functions $y = x$ and $y = 10^{x/10}$.

Graph of y=x and y=10^(x/10)

From which you can see that there are only two solutions.

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I've found that simply drawing graphs is a very good way to solve equations, if you don't need perfect precision. The error introduced this way often is small enough for applied physics, e.g. – Jens Aug 3 '10 at 9:37
After looking at the graph, you could study the equation analitically, of course (as Qiaochu Yuan did). – zar Aug 3 '10 at 15:33
ImageShack seems to have deleted your image, and replaced it with an ad banner. If you can, please reupload the image (or something equivalent) using the image upload button in the editor toolbar (which will upload it to Stack Exchange's imgur account). – Ilmari Karonen Aug 17 '15 at 16:17
Seems good now, even if I didn't reupload anything. Can you see the image now? – zar Aug 21 '15 at 20:32

Open a graphic calculator such as
Enter a function:
y(x) = 10^(x/10)
and another function:
y(x) = x
Click the intersection button and then the intersections on the image.

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For some meaning of "solve"… – ShreevatsaR Aug 5 '10 at 15:33

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