I just read the paper in Wikipedia [1] on scale invariance of some functions or curves $f(x)$. I have a few questions on the mathematics related to this.
In ref. [1], $f(x)$ is said to be scale invariant when
$f(\lambda x) = \lambda^{\Delta} f(x), \ \ \ \ \ \ \ \ \ \ \ \ Eq. [1]$
where $\lambda$ is a scale factor, and $\Delta$ an arbitrary exponent (they didn't provide the definition domains of such constants).
1) Is it the most formal definition of a scale-invariant function ? What are the definition domains of $\lambda$ and $\Delta$ ?
I understand why the the monomials $f(x)=x^n$ is scale invariant but what about
2) the logarithmic function $f(x) = \mathrm{ln}(x)$?
3) And the spiral logarithmic function defined as
$\theta = \frac{1}{b} \mathrm{ln}(r/a)$ ?
I have some difficulties to prove that the above functions are scale-invariant according to Eq. [1], typically because of Eq. [1], I have a sum of logs and I can not factorize. I am probably missing a property on logarithmic functions that could help...