# How to solve for the equation $ax \exp(bx)=c$?

How to solve for the equation $ax \exp(bx)=c$? It is known that $x\geq 0$.

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## 1 Answer

Solving this equation requires the Lambert W-function, which when applied to c gives the solution for a=b=1. Barring trivial cases, I don't think there's any easier way to derive it.

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In general, we can rewrite the equation as $(bx) \exp(bx) = \frac{bc}{a} = d$ (say). The solution for this is $bx = W(d)$, or $x = \frac{1}{b}W(d)$. – Srivatsan Nov 23 '11 at 22:55
Srivatsan's comment ought to be the answer, methinks. – J. M. Nov 24 '11 at 8:33
Thanks, it is helpful. And what about more general equation ax*exp(bx+p)=c? And how to find a solution for W(d), if d>=0 and is supposed to be a real number? – Sergei Nov 25 '11 at 0:14
axexp(bx)*exp(p)=c, then bxexp(bx)=b*(c/a)*exp(-p)=d. Am I right? Let d = 5. How to find the value of W(d)? Taylor series expansion around 0? – Sergei Nov 25 '11 at 0:33
@Sergei: Why not ask a new question? – J. M. Dec 4 '11 at 4:54