Does a logarithmic branch point imply logarithmic behavior? The complex logarithm $L(z)$ is given by $$L(z)=\ln(r)+i\theta$$ where $z=re^{i\theta}$ and $\ln(x)$ is the real natural logarithm. It is well known that $L(z)$ then sends each $z$ to infinitely many values, each of which are different by an integer multiple of $2\pi i$. If we imagine going along the unit circle in the $z$-plane starting from $1$ and working anti-clockwise, the image of it under $L(z)$ will be a segment of the vertical line $i\theta$ with $\theta$ a real number. However we clearly see how $1$ has mapped to $2$ different values, and how we could continue on mapping it to infinitely many values by winding around $0$ indefinitely. We then call $0$ a Logarithmic branch point of the mapping.
My question is, do there exist mappings that have logarithmic branch points that themselves have nothing to do with logarithms i.e. the logarithm does not appear explicitly somewhere in it's definition or it does not behave like a logarithm. Does a logarithmic branch point indicate logarithmic behavior?

I think I may have an answer;
$f(z)=z^p$ with $p$ an irrational number. If $p$ is rational, then the order of the branch point $f(z)$ at $z=0$ is just the denominator in lowest terms. So if we take the limit as $p$ goes to an irrational number, we should get an infinite order branch point.
