How do we solve $xa^x = b$ for $x$?

Question

Consider $x9^x = \frac{3}{2}$. I used wolfram alpha and found its solution to be $x = \frac{1}{2}$. But I don't know the method to get this solution. So, how do we go about solving this problem, mainly the general problem: $xa^x = b$?

Answer 1

This problem is made for the Lambert W function which is defined so that $$W(xe^x)=x$$ There a few standard manipulations to get equations into this standard form. In this case we have $$xa^x=b$$ $$x e^{x \ln a}=b$$ $$(x \ln a) e^{x \ln a}=b \ln a$$ $$W((x \ln a) e^{x \ln a})=W(b \ln a)$$ $$x \ln a=W(b \ln a)$$ $$x =\frac{W(b \ln a)}{\ln a}$$