If the equation is
$$\frac{p^{x}}{x} = c$$
where $p$ and $c$ are constants.
then find $x$.
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1What are $p,c$ and $x$ ? – Harsh Kumar Apr 09 '17 at 10:52
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@HarshKumar p and c are both constants. x is the variable. – Amit Hasan Apr 09 '17 at 10:54
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4Using Lambert $W$ function, $$x=-\frac{1}{\ln p} W\left( -\frac{\ln p}{c} \right)$$ – Ng Chung Tak Apr 09 '17 at 10:58
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2Try to remember (since you will need it quite many times) that any equation which can write or rewriye $A+Bx+C\log(D+Ex)=0$ has solutions(s) in terms of Lambert function. – Claude Leibovici Apr 09 '17 at 13:13
1 Answers
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There does not exist an elementary closed form solution. However, one can express it in terms of the Lambert W function.
By definition, the Lambert W function is the inverse of: $$f(z)=ze^z \tag{1}$$
Firstly, we use the identity $p^x\equiv e^{\ln{p}\cdot x}$. This gives: $$\frac{1}{x}\cdot e^{\ln{p}\cdot x}=c$$ Now, let's substitute $u=-\ln{p}\cdot x$. Therefore: $$-\frac{\ln{p}}{u}\cdot e^{-u}=c$$ After some algebraic manipulation, we obtain: $$ue^u=-\frac{\ln{p}}{c}$$ Now, we may use the definition given on $(1)$. Thus, we obtain: $$u=W\left(-\frac{\ln{p}}{c}\right)$$ Substituting back for $u$ gives an explicit solution for $x$.
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