Scale invariance and the Mellin transform? The Mellin transform of a function is given by:

$$\mathcal{M}[f](s) = \int_0^{\infty}x^{s-1}f(x)dx$$

Supposedly, the magnitude of the Mellin transform is invariant to scaling, analogous to how the magnitude of the Fourier transform is invariant to translation. Yet, when I try to derive this I get:

$g(x) = f(kx)$

$|\mathcal{M}[g](s)| = |\int_0^{\infty}x^{s-1}f(kx)dx|=|\int_0^{\infty}(\frac{x'}{k})^{s-1}f(x')\frac{dx'}{k}|=|k^{-s}||\mathcal{M}[f](s)|$

which is not invariant, though it does have a simple relation between them. Where did I go wrong?
$k^{-s}$ is proportional to $s$ coefficient so you can extract and compare $k$ by $\log \dfrac{g(s)}{f(s)}$ for each integral of $s$ like you do with shift on fourier.
$k^{-s}$ is proportional to $s$ coefficient so you can extract and compare $k$ by $\log \dfrac{g(s)}{f(s)}$ for each integral of $s$ like you do with shift on fourier.