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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
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Besides, that's not a complete answer. x ≈ 1.37129 works as well. –  Ben Alpert Aug 3 '10 at 3:18
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@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
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@Kaestur probably we should give OP the benefit of the doubt for now, then –  Grigory M Aug 3 '10 at 8:01
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3 Answers

up vote 17 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
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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
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You can study and graph the two functions y = x and y = 10x/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
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After looking at the graph, you could study the equation analitically, of course (as Qiaochu Yuan did). –  zar Aug 3 '10 at 15:33
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Open a graphic calculator such as fooplot.com
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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