solving for $f(x)$:$\frac{d \ln f (x)}{d \ln x} = \frac{\alpha}{x}$

Question

I have a continuously differentiable function $f(x)$ that satisfies $\frac{d \ln f (x)}{d \ln x} = \frac{\alpha}{x}$, with $f(0)=0$. How can I find $f(x)$?

Answer 1

The question is equivalent to finding $f$ s.t. $\frac{f'/f}{1/x}=a/x$ given the initial condition. Solve the ODE.

Answer 2

Hint: $$\frac{\mathrm{d}\ln[f(x)]}{\mathrm{d}\ln(x)}=\frac{\frac{\mathrm{d}\ln[f(x)]}{\mathrm{d}x}}{\frac{\mathrm{d}\ln(x)}{\mathrm{d}x}}=x\frac{\mathrm{d}\ln[f(x)]}{\mathrm{d}x}.$$