Find $f(x)$ for $f'(x) = f(x) \ln(f(x))$ and $f(0) = 1$ $$f'(x) = f(x) \ln\big(f(x)\big)$$
$$f(0) = 1, \qquad  f(x) > 0$$
I am studying for finals on my own and this exercise is really bothering me because I can't seem to solve it. If I divide by $f(x)$ I get $\ln'f(x) = \ln f(x)$ but then I don't know what to do. Any help is appreciated! 

writing your equations as $$\frac{dy}{dx} = y \ln y $$ you can see that it is separable. so $$\frac{dy}{y\ln y} = dx \to x =  \int_1^y \frac{dy}{y \ln y} = \int_1^y \frac{d \,( \ln y)}{\ln y} = \ln (\ln y)\big|_1^y = diverges$$
there is trouble with the initial value $f = 1$ at $x = 0.$
