I found the following definition of "scale invariance" in the book Critical Phenomena in Natural Sciences by Didier Sornette.
A function $f$ is scale invariant if there is a number $\mu(\lambda)$ such that $f(x)=\mu f(\lambda x)$ for arbitrary $x$ and $\lambda > 0$.
However, I also found a similar but different definition in Wikipedia:
The requirement for $f(x)$ to be invariant under all rescalings is usually taken to be $f(\lambda x) = \lambda^\Delta f(x)$ for some choice of exponent $\Delta$, and for all dilations $\lambda$.
I can understand the geometrical motivation of the first definition of "scale invariance". However, in the second definition, it is somewhat curious why the scaling factor in the right hand side should have the form $\lambda^\Delta$. I can easily show the second definition implies the first one. I tried to derive the second definition from the first one, but failed. I'd like to know whether the two definitions are equivalent or not.
Using the first definition repeatedly you can get $$f(\lambda^nx) = \mu f(\lambda^{n-1}x) =\dots = \mu^n f(x)$$ for all $n\in\mathbb N$. Hence, $\mu(\lambda^n) = \mu(\lambda)^n$ holds. Applying this to $\lambda^{1/m}$, one can conclude with $$\mu(\lambda^{r}) = \mu(\lambda)^{r}\tag{1}$$ for all positive rationals $r$.
In principle, one can imagine (1) that holds for positive rationals, yet fails for some positive irrational number. But any example would have to be very artificial. This is why it is not significant loss of generality to assume that (1) holds for all positive $r$. And this leads to the second definition $\mu(\lambda)=\lambda^\Delta$, with $\Delta = \log(\mu(2))/\log(2)$.