Are there any other methods of solving equations of $ax = b \ln x $ form, or is iteration the only approach worth trying? (We now strictly suppose that $a, b \neq 0 $).
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You mean exact solution or numerical approximation of a solution? – user37238 Sep 02 '13 at 12:52
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@user37238, rather an approach with the usual steps when facing equations like this. – András Hummer Sep 02 '13 at 12:58
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1You can always study the function $x\mapsto ax-b\ln x$ to find its root(s). – user37238 Sep 02 '13 at 13:00
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How should I start that? I'd like to know if it can be done with simply a pen and a piece of paper. – András Hummer Sep 02 '13 at 13:04
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1You can use the Lambert W function to find a closed form solution. See here. – Mhenni Benghorbal Sep 02 '13 at 13:05
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A related technique. You can have the solution
$$ x=-\frac{b}{a} W_k \left( -{\frac {a}{b}} \right), $$
where $ W_k(x) $ is the Lambert $W$ function.
Mhenni Benghorbal
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Thank you Mhenni and thank you too, @user37238. The Lambert W function is generically a great tool for these kind of problems of mine. – András Hummer Sep 02 '13 at 13:14
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