Taking inspiration from the fact every math textbook in existence puts a picture of a nautilus shell on its logarithms chapter, I did some research, and found that many different spirals (most easily defined in polar coordinates) had a $\ln()$ term in their arc length (that is, once the integral was evaluated).
I kept trying to play with some functions to see if their arc length was just $\ln(\theta)$ or $\ln(\theta)+\text{some constant}$ or something. This is because for an unrelated project, I will need to draw a logarithmic scale (of any base) and it would be cool to spiral up a piece of paper and draw lines along certain angles and unrolling a precise log scale. (If you're wondering why I'm not printing it, it's because I may not be using paper that my printer can handle, such as cardboard)
So, I'm hoping somebody has already stumbled onto such a result and could maybe lead me in the right direction.
Thanks!