# How to figure out answer to logarithm

mycrafish
x*log(x)=0.1*x^2

My guess is that using Newton's Method would be the easiest. I don't see any easy way to isolate x in that equation.

NascentOxygen
Staff Emeritus
x*log(x)=0.1*x^2
Is log = log10? Then I'd guess about, oh, let's see, .... 10?

HallsofIvy
First, of course, x= 0 is not a solution because log(0) is not defined. So you can divide both sides by x to get log(x)= 0.1 x. You can now write this as $x= e^{0.1x}$. If you let y= -0.1x, that becomes $-y/0.1= e^{-y}$. Now multiply on both sides by $-0.1e^y$ to get $ye^y= -0.1$.
(The Lambert W function is defined as the inverse function to $f(x)= xe^x$. It is also known as the "ProductLog" function. Mathematica evaluates that function and it can be evaluated at http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ProductLog. That's what I used to get the value above.)