Can I simplify $x=\frac{\ln(|\mathcal F((\mathcal F^{-1}(\sqrt{(f^{-5/3})}))^{2})|)}{\ln(f)}$
Here is a potential starting point: it is considered  that when a function behaves as a power law  ($x(t) \sim t^\alpha, t&gt; 0$), then its magnitude spectrum as well, with $|X(f)| \sim f^{-(1+\alpha)}$. One can see for instance:

Computing Fourier transform of power law
An Interesting Fourier Transform - 1/f Noise
Sketches of a &quot;proof&quot; and references (since there are  convergence issues) are provided in an answer to I cannot find reference (paper) of this relation $u(t)t^{α} ↔^{FT} f^{-(α+1)}$.

If I plug that in your case, since $|X(f)|^2 \sim f^{-5/3}$, so we seek $-5/6=-(1+\alpha)$. I obtain $\alpha = -\frac{1}{6}$. So you would have $x^2(t) \sim t^{2\alpha}$, with a magnitude spectrum $|\chi(f)|$ in $f^{-(1-\frac{1}{3})}$, or $f^{-\frac{2}{3}}$.
Now, is the discrete signal sufficiently well-sampled and clean to verify it in the discrete domain is another question.